What’s The Math Made of, Ding-Dong?

by Belle Waring on January 26, 2014

ETA 24h later: I told my girls that I was wrong and that everyone on the whole internet explained that they could perfectly well go on and win the Fields Medal if they were inclined to be mathematicians, and that being super-fast at mental arithmetic as a child isn’t the same as going on to make interesting discoveries in math as an adult, and that I was a jerk, and also wrong. Additionally, wrong. So if Zoë (12) wants to take time out from her current project of teaching herself Japanese, or Violet (9) wishes to take a break from her 150-page novel about the adventures of apprentice witch Skyla Cartwheel, then, in the hypothetical words of the Funky Four Plus One: “They could be the joint.” [Listen to this song because it’s the joint.]

“Y’all’s fakes!”

If you’re impatient you can skip ahead to 3:20 or so. Tl;dw: the overly scientific Princess Bubblegum, having snuck into Wizard City dressed in wizard gear along with Finn and Jake, is buying a spell from a head shop place that sells potions and spells and all that schwazaa. But she wants to know what the spell’s made of. “Magic?” Then she asks…read the post title. Then they get busted.

“So, kiddos,” I asked my kids in the elevator on the way down to the pools today, “are numbers real, or are they just something people made up?” Violet: “Real.” Zoë: “Real.” “That’s correct! Numbers are real! Like what if there were a sakura with its five petals, and it were pink, but no humans existed. Would it still be pink? Would it still have five petals?” [At approximately floor 14 I decided to bracket color problems.] “Yep.” “And things that are true about the number five, would they still be true too, like would five times five equal twenty-five and stuff?” “Totally.” “Could two plus two ever equal five, if there were no people around to check?” Zoë: “No, obviously not. Even now, people have lots of different languages, but if they have a word for five, then that word is about something that’s not two plus two, and it’s twenty-five if you multiply it by itself, and stuff like that. And people discovered zero two times.” “Correct! Math is real!” Zoë: “Also people discover important things about astrophysics with math, and then the same numbers keep turning up, and why would it be like that if there wasn’t really math?” “OK, so, we can keep discovering new things about math, right?” Girls: “Sure. Mathematicians can.” Me: “Maybe you! No, not you. I’m sorry.” Zoë: “I know.” Violet: “What?!” Me: “No, you’re both very intelligent children, you can learn calculus just as well as anyone, but if you were going to be an incredible math genius or something we’d kind of already know. Sorry.” [John was doing laps at this point. I’m not sure he approves of my negative pedagogical methods.] Zoë: “What’s set theory?” Me: “It’s just what it sounds like. There are sets of numbers, right, like all the prime numbers, all the way to infinity? Theories about that.” Violet: “I’m going swimming with daddy.” Me: “OK, there’s just more math out there, waiting to be discovered—but sometimes mathematicians come up with stuff that’s crazy. Like string theory. Which maybe isn’t a theory?” Zoë: “Why not?” Me: “I think they might not have any tests at all proposed by which to prove their hypotheses.” Zoë was very indignant: “That’s not a theory at all! What is that? Me: “Math that’s really fun and weird and entertaining if you understand it? John, can string theorists not propose any test whatsoever that would prove their hypotheses or is it rather the case that we lack the capacity to perform the tests that would figure it out?” John: “It’s an important distinction and I think it’s the latter. Like, was there an even or an odd number of hairs on Zoë’s head on March 23, 2006? There’s some true fact of the matter, but it’s indeterminable.” Me: “Well they can’t be demanding time travel, Jesus.” Violet: “We should have counted!” BEST. SUGGESTION. ERVER!1

OK, so, I’m a Platonist about math. Like lots of mathematicians I knew in grad school, actually, but not by any means all. In fact, some were a little embarrassed about their Platonism. My algebraic topologist friend was of the ‘numbers are the product of human intelligence’ school (N.B. while I understood vaguely what my HS friend who was also at Berkeley did set theory was writing is his diss on, in a kind of babified ‘along these lines’ way, I genuinely could not understand at all what my algebraic topology friend was doing. What, even?) This reminds me of an idiotic discussion I had in a Classics seminar with me vs. an entire group of people (including my dissertation adviser). They all maintained that there were no structures absent human recognition/simultaneous creation of the structures. As in, absent the evolution of humans on the earth, there would be no regular geometric structures. I was just like:?! Crystals that are even now locked in the earth inside geodes, where they will never be seen? Beehives? Wait, are these all imperfect and gently irregular, and thus unsatisfactory? They shouldn’t be because many of the crystals are perfectly regular. Anyway OH HAI ITS BENZENE? I…was neither presented with any compelling counter-arguments nor was I winning the argument. It was very irritating. Then I brought up my own objection—this is steel-manning, I guess: benzene was created/isolated by humans? Like Faraday even? Fine, NOBLE GASES! NOBLE GAS MATRIXES! I can draw argon on the board! Look at how this shell is so full of electrons mmmmm this probably doesn’t want to react with anything cuz it’s so lazy amirite guys (but we can make it (but also in the Crab Nebula it’s happening naturally!) but that’s irrelevant))! I still…did not win the argument. We were forcibly moved on to another topic.

I know people wanted to discuss the external reality/human-created nature of numbers and math in the earlier thread, but we got trolled by someone who was ‘just askin’ questions’ and said I ‘had to check with each and every commenter about exactly what he/she intended’ before taking offense ever at something, say, sexist that someone said. (HhHHmmmyoursuggestionfascina—NO.) Now’s your chance!
N.B. Long-time CT commenter Z alone is permitted to use humorous quotes from recalled Barbie and Malibu Stacey dolls in his discussion with me. If anyone else does I will smite you. With smiting.

Also: don’t worry about benzene. It has been around a long, long time. Certainly since the dinosaurs started fermenting into petrochemicals. And even before that. It’s a has-been for eons, even if it hasn’t been seen.

Pedagogically, I can think of at least one tenured mathematician who’s done at least some original research of my acquaintance who didn’t grow up as a math prodigy. I mean, odds are your girls aren’t going to be mathematicians, most people aren’t. But I wouldn’t rule anything out.

They all maintained that there were no structures absent human recognition/simultaneous creation of the structures.

Pre- or post- Sokal hoax?

Not that Sokal’s stunt proved anything about the existence or not of geometric structures, but I seem to recall that it instigated something of sea change in parts of academia. Pre-hoax, a species of (usually over stated and under argued) antirealism was more or less the default official position in large swaths of the humanities, including many areas that were not directly concerned with realism/antirealism as such.

Also: if one of the facets of something being “real” is that it can be a cause with effects, then numbers are surely real, as are other less constrained forms of language. I used to be a reductionist — that the universe is matter + energy — but there are clear causal effects of language (including mathematical language) that can’t be reduced to brain circuits. The movements that lead to the typing out of comments in this thread, for example.

I’m sure some of the philosophers here could put this better and tell me what I’m talking about.

So most people who know me will agree that I’m an idiot, but it seems pretty clear to me that there are no numbers past the number one. You can obviously have different multiples of one, and name each one I dunno something like two or three, but that’s basically all trickery.

Five petals? Pshaw, nonsense. It’s 1 or nothing, which I guess means we live in a digital universe. And anyway, you have to exist at our scale to see five petals on the flower instead of a whole mist of loosely-connected molecules that you could drive a bus through, all of them made up of more 1s

And that’s the most coherent my argument gets, I’m afraid. I have literally no math or philosophy chops beyond the pshaw.

A different but related question: this has been bugging me for a long time (years), at last here’s a thread I can ask someone.

When it comes to 3d geometric shapes, scale matters, right? The relationship of volume to surface area changes with scale? So (say) a cube that’s 1cm on each side has proportionally a greater surface area than (say) a cube 1m on each side?

So a platonic ideal of (say) a cube cannot exist, because without actual physical dimensions it’s not really a cube? Because it doesn’t have a relationship of volume to surface area? Unless you express this relationship in an equation, but isn’t that cheating?

You should’ve flipped it on them: But would there even be human beings if there were no numbers in the first place? Or would there even be any beings whatsoever at all? Respect your numbers, puny humans! Number is the cause of being!

What does Platonism about math mean? Is my idea of Platonism screwed up because one of the first philosophy courses I took was history of philosophy, and before the post-epistemological turn, meaning I’m hopeless? I’ll happily say there’s a fact of the matter about 2+2=5 and 1001=1000+1 and right triangles, but I’m dubious about 2 existing in the Mind of God.

Since I’m already offending Belle, I’ll finally work up my curiosity to ask if she went to Columbia or Barnard (since we were there about the same time but IIRC they have separate departments in both philosophy and classics).

That is very particular and even peculiar definition of Real. Or ‘facet of being real’. I’m not sure how you came by it, but if you leave out the facet stuff, it’s almost scholastic. There’s still a lot of work to be done to get from ‘languages are useful’ to ‘five is prime and also real’.

This science radio show I listened to interviewed child development experts and anthropologists. In it, they said that babies (and people in societies which do not use numbers) naturally think logarithmically. And the only reason we use base-10 integers is that our parents count to us as children and we eventually just accept to this different, socially acceptable math. For me, it was *mind blown*.

Uhh, no. There is no “cat” without language (defined broadly perhaps including signs), nor is there a cat. The Reality we can talk about or think about is socially constructed. What remains is Dasein or Dass-sein.

This “science” or “mathematics” stuff is simple pure idolatry needing a hammer of philosophy.

I take objectification or Platonism or reification very seriously indeed, because I don’t see a very big ontological difference or gap between “5” and “the eternal unchanging essence of feminity” or “let me tell you the Real You.” All sin starts with Ideas of the other.

Does this make me a materialist? The material world disappears when we deny the immaterial.

Just when I get comfortable in thinking of math as a purely mental construct, they pull me back in. More seriously, Meillassoux is interesting on this sort of thing, particularly his conceptions of ancestrality, the arche fossil, correlationism, etc.

For these reasons, Meillassoux rejects Kant’s so-called Copernican Revolution in philosophy. Since Kant makes the world dependent on the conditions by which humans observe it, Meillassoux accuses Kant of a “Ptolemaic Counter-Revolution.”

This is the leap that makes metaphysics.

Kant does no such thing. He makes the world we can speak of or talk about dependent on categories etc. He says nothing about the “world” because nothing can be said.

When it comes to 3d geometric shapes, scale matters, right? The relationship of volume to surface area changes with scale? So (say) a cube that’s 1cm on each side has proportionally a greater surface area than (say) a cube 1m on each side?

So a platonic ideal of (say) a cube cannot exist, because without actual physical dimensions it’s not really a cube? Because it doesn’t have a relationship of volume to surface area? Unless you express this relationship in an equation, but isn’t that cheating?

Any real object is only approximately a cube, so when we talk about cubes I’d say we’re invoking the platonic ideal of all real objects that have seemingly equal sides and 90 degree angles.

Also, you can’t talk about relationships between surface area and volume without using equations. Any object will have its surface area increase slower than its volume if you increase the size. What’s special about cubes is that the ratio of surface area to volume is 6/s ( = 6s^2/s^3, where s is the length of the side of the cube.)

There’s a series of Eureka! objects or constructs I dunno what, where the pomposity serious guys are all “We have found the building blocks of everything!”
Then pause, then oh.
So gluons and muons and so ons.
Quark! Then…

Now there’s Penrosian quanta making the brain happen. Microtubules something something.
And a mystery bit coming in from the sun that we didn’t know was there, but it was, all along.
It was molecules for a while way back when. Greek atomic theories of the BCE.

From miasma to bacteria to viruses to prions to “Nothing more to see here, just little bricks and lego-like-things.”
From which the illusion of individual consciousness is built.
Or not. quite. there. yet.
The micro-boundaries of our living being keep getting pushed a little further in and down.

The awesome to me but evidently awful to many idea that it just keeps going.
With fun portals to other dimensions becoming available, possibly soon.

Math guys of a certain stripe evidently hate them infinites, understandably. Lots of doctorish professionals don’t like metaphysics for pretty much the same reason.

Germ theory or some more recent analog helps a lot for genetic engineering and all that, but it’s a mechanical not a wholistic understanding, and it is not the final complete and total what-is-it.

Math is like the skeleton of the thing, it is not the thing.
Skeletons are very important, essential to what we are, but they are not what we are.

Personally, I’m considering the question of the independent existence of integers to be an open one until we meet E.T.s. If the kids from proxima centauri have something recognizable as the Peano Axioms, then yeah, there’s probably something real casting a shadow on the wall that looks like “1, 2, 3…” If not, oops, we were actually looking at a mirror and telling ourselves it was a window. Again.

(Of all of my closely held and cherished crank-like beliefs, this is by far the most cherished and crankiest.)

where she distinguishes fast and slow mathematical talent and comments

“Think of it, your slowness, or lack of quickness, as a style thing but not as a shortcoming. Why style? Over the years I’ve found that slow mathematicians have a different thing to offer than fast mathematicians… One thing that’s incredibly annoying about this concept of “fastness” when it comes to solving math problems is that, as a high school kid, you’re surrounded by math competitions, which all kind of suck. They make it seem like, to be “good” at math, you have to be fast. That’s really just not true once you grow up and start doing grownup math. In reality, mostly of being good at math is really about how much you want to spend your time doing math.”

More recently on her blog, she also has a lucid demolition of the infinite series “result” in the earlier mathematics post on CT.

Like, was there an even or an odd number of hairs on Zoë’s head on March 23, 2006? There’s some true fact of the matter, but it’s indeterminable.

Pretty sure John is wrong here. Even if we narrow the time to an instant, we’ll still have to come up with criteria for whether given hairs are on a given head. (And are we counting peach fuzz? Eyelashes?)

Seriously, as you note with the hexagons of the honeycomb, reality is messy. Even the carbon atoms of the benzene ring form a hexagon only in a sense that needs some cautious defining.

Belle Waring: I’m curious, can you recall the reason that a room full of classicists were arguing over whether geometric structures exist without being “created” by human recognition?

I was sort of trolling with my Sokal remark, but it does strike me as a very 90s thing that people in all sorts of far flung areas of the humanities were getting into heated arguments over these “How do you know you’re not a brain in vat?” type of issues. Was that the time period in question?

#24- None of us exactly see why the macro past is unique but the macro future isn’t. The basic form of quantum field theory is time reversible (ok, CPT, to be precise). Nevertheless, in all our experience that irreversibility is just a fact. We can tie it in with other facts, in particular the second law of thermodynamics, but at best that just makes one big mystery out of a bunch of little ones.
Maybe there’s some anthropic argument, along the lines that Boltzmann suggested long ago. Out of some swirling multitude of existence, only the strongly T-asymmetric parts support certain types of information processing (us). But that’s not exactly settled science.

Surely there’s a position between mathematics being prior to everything and math being a matter of arbitrary human whim. Math could be like Physics–it’s a set of principles that holds true in this world, but could be different in some other world.

I think some people with more realist views of mathematics than that might call themselves non-platonist. Like, how real do you think an incomputable “real number” is? Does all the stuff inside a set with cardinality greater than the continuum actually “exist” in a meaningful way?

There are logical systems in which what’s “true” changes over time, e.g. linear logic. Of course, that just pushes the permanence one level up–some truths may change over time, so you describe timeless rules by which they change. But two can play at this game–I describe the rules by which your “timeless rules” change. That game can go back and forth indefinitely.

This actually reminds me of the Holbo post from, last summer?, on utilitarianism where the thread turned into one long back and forth on moral objectivity (well, mostly one long forth against). I’m sort of curious on what the overlap is (if any) of people who are diehard subjectivists in ethics but Platonists about math. Because that is weird, no?

(For what it’s worth, I’m a quasi-Platonist about both, but only because I’m sort of an idealist about everything. No, really.)

I get the impression that on a day-to-day basis, mathematicians, who should care about this sort of foundational stuff, don’t, in the same way generation after generation of them didn’t care whether the foundations of the calculus were solid enough to skate on, because RESULTS!.

I’m usually not a fan of the cop-out declaration that an old philosophical question is a “pseudo-problem,” but I can’t for the life of me convince myself that the question “are numbers real?” has a clear meaning or whether the commitment either way leads to interesting further commitments or consequences.

“Like what if there were a sakura with its five petals, and it were pink, but no humans existed. Would it still be pink? Would it still have five petals?”

Would it be different or less effective an argument to say this just as well demonstrates that words are real, since “sakura” and “pink” and “petals” exist even if there are no humans? It strikes me as equally strange an argument. Aren’t numbers just a more precise and abstract form of language or signs? What does it mean to ask if symbols, signs, or language are “real”?

At best it demonstrates that number-able or distinguishable things exist. And distinguishability doesn’t demonstrate anything about number as a “reality”: I distinguish a face in a newspaper photograph, but the photograph has no “real” number–it’s many dots.

Does being a realist about number commit one to realism generally or being an anti-realist about numbers commit one to anti-realism generally? What are the stakes of such a debate? What do we lose if numbers aren’t real? Not even flower petals, I’d say.

I used to care about this question when I was 25. I am now 50, and I long ago realised that it doesn’t matter. If the difference between idealist and platonic conceptions of mathematics made any difference in the world, then we would have jumped one way or another long ago. It is like arguing about the superiority of Pascal over C – though I am loth to give up König’s lemma.

The “are numbers real” question is like a stand-in for “are mathematical principles, formulas and artifacts things which exist in the real world, or are they human constructs” question, which is important since math is the language of science / all kinds of ish is predicted mathematically before confirmed empirically => if math isn’t “real” then what can we say is actually “real”?

The OP brings this all up, incidentally, pretty seamlessly going from “oh hai flower petals” to the “unreasonable effectiveness of Zoë” point (yes I copy-pasted the name because I don’t know how to do umlauts) to BENZENE RINGS WAKE UP SHEEPLE.

When I do maths I do it as an idealist – as if I was looking to discover something that exists independently of us. But my firm belief is that maths are socially constructed just like all human intellectual endeavor. Now this is not to say that in a different society, two flowers with 5 petals would make 11 petals total!

But maths is more than this: it is the practice of studying structure, counting, measurement – and this practice is historically and socially developed. For example for a long time in the mathematical practice that grew out of that of the Hellenistic world, “numbers” as in “five flower petals” where considered distinct from “numbers” as in “the length of this side of a triangle is five” – there was no concept of ‘number’ as we use now that can be used to either count or measure. This concept is a human creation, even if the flower petals are obviously not.

Similarly, the maths we practice as research mathematicians today – the centrality of the notion of mathematical proof in particular – again is very specifically historically developed out of the Hellenistic – Arabic mathematics, and in its current form is very modern; its spread around the world was a consequence of Western colonialism, missionary work (Jesuits brought Euclid’s Elements to China after a Ming Emperor appointed a Jesuit monk chief of the Imperial Office of Astronomy) and imperialism. Traditional Chinese mathematics for example considered proofs an affront to concision and unnecessary luxury; in Japan, Western-style math was introduced all at once during the Meiji restoration as part of the program of Westernization.

I would second Lizardbreath, I don’t think it can be ruled out (assuming they are intelligent) at an early age. Nor do signs of earlier hypercompetence mean very high probability of becoming one. The skill and interest sets involved change with advancement through the different levels of study. And in fact very high competence in some skill can lead to pedagogical deadends. I have an extraordinarly good intuitive graphical feeling for some types of math/physics. Yet I could never be a mathematician, reading theorems/lemmas/corollarires cause my eyes to glaze over.

MG @13, thats very interesting. A little hard for me to believe however. I’m just old enough that calculating with log tables was still taught in school -and my dad taught me years before I encountered them in school. But, my memory informs me, I was the only person in the entire school system who wasn’t baffled by the concept. And the difficulty of getting nontechnical people to accept stuff like LOG linear plots seems to confirm my suspicion, that logarithms and human brains are not natural partners.

The OP kind of reminds me of the ever exasperating “if a tree falls in the forest” we were subjected to. I’m glad that issue is now settled, as we have a humorous TV advertisement which shows an unobserved tree indeed making noise.

@Anon, #29: At best it demonstrates that number-able or distinguishable things exist.

Also innumerable things, like real numbers. Unless there’s a way to say that the natural numbers are different from the real numbers. I haven’t thought about this, aside from reading Quine long ago, but now I’m wondering whether numbers being “real” implies that the universe holds an uncountably infinite number of things and whether there’s any reason to say that numbers came into existence only when the universe did. (Not wondering very hard, though.)

No, you’re both very intelligent children, you can learn calculus just as well as anyone, but if you were going to be an incredible math genius or something we’d kind of already know. Sorry.

Srsly, Belle? Childhood precociousness has very little to do with the ability to make real progress on things. The smartest, olympiad-winning-type math-geek kids I met when I was in high school are now working in finance and destroying the world economy or something. Some people who got a late start but are good at thinking slowly and carefully about things manage to outstrip the fast-out-of-the-gate types. Aren’t girls going to get enough cultural messages about what they can and can’t do already, without the added message that if they haven’t displayed Ramanujanian genius at an early age they’ll never amount to anything?

And anyway, one can be a mathematician and discover new things about math without being an “incredible math genius.” Most people are just muddling through their career making incremental progress, and there’s nothing wrong with that, is there? A decade-ish into my academic career I’m pretty much resigned to never doing anything memorable, but it’s a living.

Perhaps “mathematics is real” is just a translation of “science is not arbitrary”.

Suppose theory A predicts observation X and theory B predicts observation Y. I do the experiment and observe Y. Sounds like a good case for B, but the former proponents of A amend their theory and now insist that X will always be observed except for that one experiment I just did in which Y occurred. And they keep doing this no matter how many experiments we do.

Intuitively, I think there is something absurd about believing an infinite series of modified theories in which all the future predictions remain unchanged despite every failed prediction. But if mathematics is just a human invention, why should I prefer the unchanging B to the newest modification of A? If math is invented, why isn’t science arbitrary?

@ 29 – Aren’t numbers just a more precise and abstract form of language or signs?

That’s what I’m thinking. Symbols are perplexing because they are pointers, not things. The same thing that makes language useful also makes it extremely deceptive. For example, conveying a false sense of permanence and identity. But also shifting our attention from the world of experience to that gray zone of symbol manipulation we call “thought.”

Decades ago as a senior in college I took a graduate seminar in Harmonic Analysis. Within two weeks it was clear to me that I really wasn’t able to understand concepts being introduced whereas others in the class had no such problem. Since I knew a couple of the other students and hadn’t found them specially bright in conversation it caused me no end of mental anguish. To help my recovery, I decided that kind of math really was pretend and became an engineer instead.

Well, in one sense logarithmic scales are built into our sensory apparatus: we experience both sound and light via log scales. Ask someone “which light is N times as bright?” and they’ll point to a light that is putting out something-to-the-N times as much light. A light putting out four times as much light will appear, meh, maybe twice as bright. That’s also why decibels are log-scaled. (Differently put, our phenomenology is doing log compression).

But it’s a few more steps from that, to saying that we are comfortable doing conscious thinking or conscious processing via logs. That takes a bit more practice.

Whatever you think about numbers, though, you’ll have to grant that there have always been eternal, mind-independent blog-topics, which would have existed even if no internet had ever noticed them.

My somewhat wishy-washy compromise position is that numbers don’t have independent existence per se, but the fact that some aspect of reality can be described in a mathematical way is an objective eternal truth. So there is structure to the universe, but mathematics is not identical to that structure, but merely a tool humans use which allow us to study and replicate it. And pure mathematics is real in this sense too: even if there’s no physical aspect of reality that corresponds to a mathematical principle, it’s still objectively true that if someone set up an axiomatic system in such and such a way, you’d get such and such results.

Platonism, especially when you’re talking about the actual views of Plato, tends to imply all sorts of weird metaphysical baggage which seems unnecessary. Mathematics is too elaborate and creative a system for it to have been here all along. Mathematicians are constantly making up new mathematical systems, and how are we supposed to distinguish the ones which are “really real” versus more playful nonsense. But the mathematical analysis of reality is still a bit too staggeringly effective to just act like reality had absolutely nothing to do with mathematics until humans started playing around with it.

And Belle, I’m totally with the people saying you should quit telling your kids they can’t do math.

The fact that they have not already shown prodigious abilities is, I think, pretty good evidence that they will not wind up wandering the halls of Fine Tower scribbling gibberish on blackboards, or searching through the dumpsters of Telegraph Avenue.

But it says very little about whether they could have productive careers in the field of math (the very large and very disparate fields of math).

They have as good a shot at making money doing math as most people do, and it is the blameworthy ignorance to think you know otherwise.

UserGoogol 01.26.14 at 9:54 pm @ 45:‘… But the mathematical analysis of reality is still a bit too staggeringly effective to just act like reality had absolutely nothing to do with mathematics until humans started playing around with it.’

Tegmark’s theory is that the underlying reality is ‘mathematical'; when humans do mathematics, they are constructing things which approximate this fundamental reality in perhaps very simplified form. ‘This fundamental reality’ being the one in which they happen to be embedded — if you can create one universe from nothing, why not many? Why not an infinite number? Il n’y a que le premier pas qui coûte.

The quote from Cathy O’Neal @24 is important. Though I remember math competitions (admittedly at a school that was never competitive): here are a lot of tricks to memorize about Taylor series, no I don’t have any textbooks that would actually explain Taylor series, and oh yeah, the last couple of competitions didn’t involve problems from these sheets, but Euler invented this all himself from scratch when he was five years younger than you are now, you didn’t memorize enough trick sheets, and maybe you’re just not good enough for college-level math. Ugh.

The curious thing, for me, is that we would think that we’re translating logarithmic-appearing shifts in our observations – into something like once, twice, thrice. But that begs the question: How do we know we aren’t actually translating base ten language (beliefs? rubrics? schemata?) into logarithms, for the more fundamental circuitry of mind to work with?

Kids (and fans of logarithms) giving the answer that 3 is halfway between 1 and 9 seems to be one data point towards logarithms being related to a more basic mode of thought. Sure, in turn there may be base 10 or differentiable manifolds or whatever underneath that, but it looks like as far as we’ve gone that logarithmic thinking is the more basic, so the skepticism here seems kind of shaky to me.

If we posit blue numbers and red numbers, then numbers can have properties that are interesting or boring, depending on your reaction. (Blue and red numbers are in “The Education of TCMITS”.)

On a sphere, a great circle is like a line in a plane, except that if you number the points on a great circle, you can’t say that one number is greater than or less than another, since you can start from one point and reach the other by going in either direction. You can get to positive numbers by starting from a negative number and going towards and then through negative infinity, and vice versa. To get greater than and less than relations, you only have to eliminate from the great circle the point at infinity. That gives relations like ordinary numbers. But you could also eliminate the point at zero. In that case positive numbers are less than negative numbers. Do such numbers exist or not?

I’ll toss in my little stink bomb… How about the fact that the ‘N’ in the cycles of N-year cicadas are all prime numbers? Does this make primality more real? Or does it make natural selection intelligent? Or mathematical? Or what?

Um… Is nobody going to comment on how PB is becoming more and more of a personified allegory for the American police state/empire?

This detail must have somehow slipped my memory when I had watched the episode for the first time, but here she’s outright torturing the Ice King to divulge secret information held by members of a different sovereign power structure… Taken together with the development in the most recent episodes, the show’s really becoming a pretty blatant satire.

What is Platonic may just be the shaped of our cognition with respect to the universe. We evolved that way, we have these patterns inside us, so we see those patterns outside. Furthermore, those outside patterns would be here, even if we weren’t. Therefore math is a valid metaphysics, but it may not be a complete one. This could happen for example if our cognitive structures evolved along certain possibilities allowed by the universe, but not all of the possibilities. So there might be alien beings on other planets who have nothing like mathematics, but have explanatory means which also, like ours, reach far into the stuff of the universe, but because their cognitive structures evolved quite differently along other principles afforded by the universe. Both species — the humans and the aliens on the other planet — could claim that even if there were none of us remaining — humans or aliens — our respective patterns would exist in the universe. So what is Platonic to us may not be Platonic to another.

Pretty sure ‘pink’ isn’t real after once being told off by a girlfriend for mistakenly pointing her in the direction of a ‘salmon’ colored jersey instead of the indistinguishable ‘pink ‘ one she was looking for.

I think that the point is that there is a difference between ‘real’ and ‘true.
There is a reality, that is real.
We create some model to describe this reality.
Models that correctly predict this reality are true.
But these true models are not the same thing of the reality that they describe, and are necessariously very partial.
So relatonships among numbers are true, but are not real, as they represent just a very small slice of the reality they describe, a slice that we selected because subjectively more relevant or easier to model.

Bas van Fraassen somewhere invites you to suppose there are two worlds: One in which mathematicians think there aren’t numbers, but there are; and another in which mathematicians think there are numbers, but there aren’t. And they both prove exactly the same theorems.

Lee Arnold @57 (currently): I think we strongly agree here. There are probably deep structures to the universe (even if such “structures” are underneath merely statistical approximations of the range of outcomes of a bunch of ultimately chaotic behavior blah blah blah), and it’s possible that “arithmetic” is only one of a possible number of (potentially mutually contradictory or at least not obviously overlapping) means of explaining those structures.

But it’s worth noting that this is very much not the position taken by the “strong mathematical platonists”, such as Belle above. (Steven Landsburg also comes to mind as such a proponent.) The position there is that, for example, the natural numbers exist in the same way that a rock exists, because integers describe a property of the universe so fundamental that the rock could not exist without them, and any intelligent species capable of in some way describing a rock would of necessity derive the existence of the natural numbers along the way. To put it mildly, I find this undersupported by the available evidence.

e.g. “The radial pattern on the left of the plaque shows 15 lines emanating from the same origin. Fourteen of the lines have corresponding long binary numbers, which stand for the periods of pulsars, using the hydrogen spin-flip transition frequency as the unit.”

neat post. I suck at Math, and this was very educational. The sad thing is that I am pretty sure Math still exists outside my cognition, but I am basically mathematically illiterate (innumerate?) so I would have no way to prove its existence or non-existence (cant even manage much algebra). It would be like arguing about the Bible, or Plato’s Republic, if you couldn’t read. I suppose there might be a way for me to learn Math, but I’m in my forties and kind of pressed for time as it is.

I think once we borrow down to the level or particles, or “quantum” mechanics, we have to confront integers. At the microscopic level we have quantum numbers (such as the energy levels of a hydrogen atom), so integers seem to be a fundamental property of the universe.

Do you say you are a Platonist about numbers because you believe in numbers as mathematical objects? Or do you go whole hog and affirm the existence of the eidetic number, which mathematicians know nothing of?

Anarcissie: The Mathematical Universe Hypothesis or other kinds of modal realism have a certain appeal to me, (by putting such a massive array of possible objects as being part of the totality of existence, you allow quite a lot to exist without “unfairly” picking and choosing) but I think the kind of raise the question “well what does it mean to exist, then?” If existence is something possessed by such a vast array of radically different entities, then it loses whatever punch it might have. In the common sense meaning of the word existence, some things don’t exist, so “existence” distinguishes those from the things that do exist. But once you expand the definition of existence to include anything that can be put into a mathematical system, saying something exists becomes kind of tautological.

Badiou wrote a book on this -“Number and numbers”. Unlike even “Being and Event” which I found hard, but doable in small chunks, this one I gave up on about 100 pages in, as I found my brain becoming number and number.

The cartoon reminded me of Clarke’s “Any sufficiently advanced technology is indistinguishable from magic.” When my “Science Fiction and Fantasy” class tries to determine the differences between the two genres I bring up that one. “What if Gandalf just uses nanotech?”

On considered reflection last night just after I put up the post I thought, I should tell my children they can be mathematicians if they want, why not? That was kind of a dickish thing to say. I was clearly being purposefully sarcastic, but still. I shouldn’t even be being purposefully snide to my children, what’s the point of that! But I already stayed up past midnight so I didn’t edit it. When the girls get home from school I’ll tell them the entire internet wants them to know they can be awesome mathematicians. And look, they can read and write Mandarin, and Zoë is teaching herself Japanese–it’s not like I think they’re stupid or anything.

am80256: It is Zoë’s considered view that “PB and the Ice King are the players, Finn and Jake are only pawns who don’t know there is a game, and Marcelline doesn’t care.”

Doctor Memory: that’s my crankish view also! It’s just that whenever sci-fi writers try to come up with aliens who use only irrational numbers all the time in ordinary life it is very…uncompelling. See: Greg Bear’s Anvil of Stars (that’s the sequel one, right?). It’s not that I think everyone would be using base ten, it’s just that I think there would be integers, and negative numbers, and mathematical operations such as addition and division, and so on.

Everybody: the math isn’t in the mind of God, necessarily, but I do think it exists independently of humans and that there’s some meaningful sense in which one can say humans discovered zero twice and not that they invented zero twice.

bianca steele: I went to Columbia for undergrad. Other Peeps: the seminar argument was at Berkeley and was indeed in the 90s, but surprisingly late in the 90s! Post Sokal hoax, even. I don’t know how the discussion even got pushed over there. No, actually, probably it was me because I thought that on that terrain I would obviously, totally prevail! Completely not-winning the argument while completely winning was frustrating.

Martin Bento: No, they’re not, but it doesn’t matter, because they are making this one. Adventure Time is the greatest story ever told. Season One is more pointlessly silly, but the later seasons get strange and psychologically interesting and moving.

Belle: if we’re ranging into the realm of “my crankish ideas projected onto the canvas of dodgy sci-fi writing”, I’ll aver that “only using irrational numbers” or whatever strikes me as a kinda cheap dodge. I have a sneaking suspicion that if we ever met spacefaring ETs, we might not recognize anything they do as “math” or even “cognition” at all. The only writer I can think of recently who tried to take the idea seriously in a “hard” SF context is Peter Watts in “Blindsight”, although Watts was more concerned with the question of consciousness rather than math qua math.

(You can download Blindsight for free from Watts’ website — I’m not linking only to avoid getting eaten by CT’s increasingly erratic anti-spambot. Google will lead you there, and you totally should.)

AFAICT the problem for mathematical “Platonists”/realists/foundationalists is this: what would legitimate such a view other than a sheer subjective conviction born out of mathematical “experience”? IOW what difference would it make, what implications would follow, what tests could be devised?

One might start with the question of whether mathematics, beyond the most elementary operations of finite counting and perhaps simple arithmetic, would be possible without symbols, those peculiar combinations of both signifieds and signifiers, which are not just a material convenience or crutch, but are required to designate the objects of thought in the first place. But symbols are the result of underlying systems or sets of intermeshed rules and rules are behavioral, enactments that constrain behaviors. And further there are plenty of valid significations that don’t have referents outside of a system of signification: for example, “truth”. Can anyone show “truth” as a designated object of reference? (Hence, in their different ways, both Heidegger and Davidson take “truth” to be primitive, not capable of further explication, but the basis of further explications, though I myself would have reservations about that). So what then is special about mathematical “truths” that their apparent intuition should be eo ipso referential?

Beyond such intuitionism, the usual claim is for the unreasonable efficacy of mathematics in formulating scientific understandings of the world. But, aside from rather fetishizing the development of mathematical physics in the 17th century, (together with the accompanying 17th century metaphysics, all of course, in a Euro-centric context), that claim is based precisely on the fact that mathematical constructs often long precede any “practical” application to scientific research, (though the case of calculus is the reverse), and confuses “pure” and applied mathematics. No doubt, beyond elementary counting operations, the impetus toward the development of “higher” mathematics had practical import in ancient civilizations: the accounting for debts and the formation of calenders. Both of which were the function of priestly castes. And, indeed, the association of math with astronomical observation, essential for both it and calender formation, gave to mathematics a mystical, heavenly cast. And the orientation toward pure mathematical formalisms has a placelessness and timelessness that lends it a mystical feel and likely was first cultivated, like music, dance, or poetry, in a cultic, religious context, “for its own sake”. My rough understanding of Plato’s view is that mathematics and “dialectics” are related by analogy. The sort of assurance that mathematicians could achieve via procedures of proof was not identical to philosophical reasoning, but philosophical reasoning could achieve the same sort of assurance, whereby, though not everyone could follow it, one could nonetheless hold fast to its insights. And, of course, the invariance and purity of thought involved in mathematical truths would form that analogical model of attaining “eternal” philosophical truths, in the forms and idea of the hyperouranios topos. But it was only in the 17th century, (with its metaphysical obsession with the intractable idea of infinity) that mathematical formalism and scientific application came to be heavily identified, (partly in an idea of “proof”, which itself was weighted with the absolutism of Christian dogmatism).

But the pure formalisms of higher mathematics are quite distinct from their technical applications in the formation of more-or-less empirical knowledge. The latter involves working through conceptually a specific domain and the application of mathematical devices to domain-specific parameters that are measurable and capable of being extended through mathematical inferences of the chosen sort. IOW no matter how mathematically formalized a scientific theory may be, the chosen formalism always encodes domain-specific concepts, even if those concepts are virtually inexpressible in other terms, (such as, purportedly, the Dirac equations describing electron orbitals in quantum mechanics). And this is possible precisely because there is a large repertoire or library of pre-existing mathematics to choose from, to suit the patterns sought and recognized and their explanatory purpose and measurable variables. Two theories in two distinct domains might have entirely homologous formalisms applied, but they are nonetheless conceptually quite distinct and have to be justified each in their own terms and contexts. But it’s not that a pre-existent mathematics somehow describes the intrinsic structure of the universe; it’s rather that we find such mathematics useful in pursuing our explanatory aims. (Though, mind, I don’t think scientific knowledge is reducible to instrumentalistic terms, nor that mere predictive success eo ipso validates such theories, nor that all valid knowledge must or even can be cast in mathematical terms).

No doubt mathematics can be “beautiful” or give a sense of harmony. The wheels whirring in our brains seeking fulfillment in pattern-recognition occur in mathematical thinking as much as anywhere, (which then get projected onto the outside world as “the music of the spheres”). And no doubt mathematical thinking in pure formal operations is a much different process (with a different neural basis) than discursive reasoning, (which is largely a reflexive, internalized form of language usage), giving rise to an illusion of “pure thought”, divorced from worldly contexts and more-or-less material contents. But precisely such metaphysical illusions and the false mastery of a transcendent knowledge that precedes and surmounts the worldly basis of knowledge should be resisted.

My bottom line here is that the discernible or useful implications cut against mathematical “Platonism”.

One might start with the question of whether mathematics, beyond the most elementary operations of finite counting and perhaps simple arithmetic, would be possible without symbols, those peculiar combinations of both signifieds and signifiers, which are not just a material convenience or crutch, but are required to designate the objects of thought in the first place.

Thing about this tho is that it’ll commit you to anti-realism about non-observational (“theoretical”) entities in science—like, I don’t know, quarks or whatever. Which obviously is a position held by many people in the area, but always seemed to me a bit drastic.

More generally, I think that at least part of what motivates Platonism about math is a recognition of the necessity of mathematical truths—their obvious non-arbitrariness. (Which of course isn’t quite the sort of motivation BW is presenting in the post—at least I think not?) It would be I think a total disaster if ‘humans invented math’ were taken in a sense whereby it implied that mathematical truths could be, e.g., changed at will. (We made it; we can make it otherwise! That obviously seems a nonstarter.) All that said, I find that there are better ways to account for the non-arbitrariness of math—its, umm, a priority—in ways other than a bald Platonism. Tho it’s not much help for the metaphysically weak-kneed.

What does “necessity” mean here, if not formal implication? And systems of rules impose their constraints on their users, while enabling those users. The case with math is not really all that different than with natural language: neither one is simply subject to our arbitrary whim. Your worry seems to stem from an implicit residual attachment to nominalism.

js: No, that was part of the point. ‘Could 2+2=5 if no humans were around to check’ is meant to ask also ‘if we stipulate that humans invented the concepts 2, addition, equality, and 5, could they refine or alter their ‘invention’ in such a way that 2+2 did, in fact, equal 5?’ Obviously, totally, self-evidently not, right? But in the case of the sakura, for example, we may create literary conventions around it, such that it represents springtime more fully than any other event/fact/growth/development naturally occurring during the period stipulated to be ‘springtime’, and is the most perfect topic for traditional seasonal poems and songs. However, Japanese culture could change in the future, so that the sakura was no longer the most popular and symbolic flower, and no one ever mused on how the bare, black branches of the cherry boughs in winter, blooming with snow, resemble the tree in springtime, heaped with blossoms that flutter to the ground as the short-lived flowers live and die almost at once, opening pinkly and swiftly and then paling as they curl and fade and drift away–snowflakes tossed on the winter wind. Yep, e’erybody could suddenly be all ‘I have zero fucks to give, cherry blossoms.’ This would not result in an inconsistency so wrong that it was impossible even to think about properly, such as 2+2=5. It would just be super-weird for Japanese people to do that.

Wait, re-reading the thread, bianca steele, why did you think you were offending me in the first place? Because if I had gone to Barnard I would have been lame because it’s so much easier to get into? I don’t even understand what you’re talking about. It’s not rude to ask people where they went to college, generally. If they choose to look into the middle distance and say “Boston” that’s all on them. At the time I had, hm, variously: long brown hair, long blond hair, 1/2-inch long brown hair, yep just kep on being short for quite a while, then started to grow out a good bit because I decided I wanted to have long hair when I was 25 (mission accomplished there.) I had black Buddy Holly glasses. Naturally. I took some classes at Barnard in Classics. Mmmm, Aristophanes, and…something. The woman who taught Aristophanes was great.

@72 – I agree with every bit of that analysis, with the exception of the Ice King being a player in the true sense. I view him more as an Id-driven nuisance, powerful enough to be taken seriously, but not really caring about the big-picture game as such any more than Marcelline. He’s not working towards improving his position, setting up buffer client states or collecting and developing weaponizable technology. In that way I see him as more analogous to adverse natural forces, appropriately enough.

Currently, PB seems to me to be the only game in town, with the Lich operating on a different level but now perhaps taking PB seriously as an obstacle to his plans.

1) It is a marvel of language that we can create expressions or statements that “represent” the impossible, unimaginable, or the unthinkable that are such because of the intrinsic or internal nature of language. These kinds of statements are not of the “there is a unicorn” kind.

2) Lord I don’t know any Cantor. But this means to me that perhaps language is of a cardinality greater than either perceived phenomena or human thought processes, and so language cannot be mapped onto, completely and reliably, and is incommensurable with either.

3) I also feel that we are falling into some weird variation of Anselm’s ontological argument here, with 1 & 2 above being a parallel attempt at a historical refutation.

Anselm defined God [mathematics] as “…that than which nothing greater [different] can be conceived,” and then argued that this being could exist in the mind.

Belle, The offense was in the other thread, that I was referring to. Re. Barnard, it’s the little twinge of snobbishness I feel when someone assumes I went to Barnard (or the Engineering School), or see “graduated from Columbia University” in the wedding announcements page and think, “I bet it wasn’t really Columbia.” There was a wee bit of tension there in 1984 or thereabouts. I do sometimes say I went to college in “New York” because nobody has heard of Columbia anyway, and it actually sounds better. And it shows you the circles I move in, but the only time I’ve ever heard the “Boston” thing was at a wedding where an in-law I’d never met but knew all about said, “I went to college in New Hampshire.” Yes, I’ve never heard of Dartmouth.

The glasses: my roommate had those. I copied her for a little while, but realized it would annoy her. I’m sure you wouldn’t remember me, I was the bad looking person hiding in Gisela Striker’s class behind a woman who’s now teaching at Harvard.

So I’m a mathematician and definitely not a Platonist, largely through experience and the fact that, on reflection, Platonism just doesn’t really make any sense to me.

For instance, I honestly don’t have an opinion on whether the Axiom of Choice is true or not. Its convenient, and thats about all I am willing to commit to. But the question of whether it is true seems somewhat incoherent.

Let me argue by analogy. Ask yourself whether “en passant” is a valid move in chess. Yes, of course it is. Now ask whether you want to include it if you re-design chess. Well, you might have an opinion on that, and thats cool. Now ask yourself, in the middle of a re-design, whether it is “true” that a pawn can capture as in “en passant” and whether your rules should therefore reflect that. Thats more or less what mathematical questions about truth look like to me – “truth” looks pretty nonsensical here. Your opinion about the validity of the rule might be so strong that you desire to claim it as “true”, but it is hard to see this as more than a rhetorical flourish. You are just pounding the table really hard while trying to convince me.

As to Belle’s question….if I interpret “+” as “add the next number plus one” then “2+2=5″. Easy, done. Conceivable. I just conceived it! And I am certain that that is absolutely not satisfying. Why? Because you want to keep the natural bits of arithmetic, whereas I just arbitrarily introduced a change (which arguably isn’t good for aesthetic reasons or whatever). But ok, but then if I keep the natural bits and the way they fit together, then saying that “2+2=4″ is just an expression of those rules and the way they fit together. It is true, in the same way the pawns may take using en passant by standard rules of chess is true. And, of course, we can imagine different ways to construct the rules of chess, just as for mathematics (although most people don’t quite realise that, but Godel is knocking around here, challenging your notions of “truth”). It is also the case that people have an intuition about arithmetic that makes it feel as if the objects are real….but I think that says far more about human psychology than anything else.

No, you’re both very intelligent children, you can learn calculus just as well as anyone, but if you were going to be an incredible math genius or something we’d kind of already know. Sorry.

Half of my Facebook friends are active mathematicians (that is, if I were on Facebook) and a number are really top class (what would be a credible measure to an outsider, oh I know, they have a Wikipedia page!). Among them, two or three were child prodigy. The rest is average chump all the way.

but sometimes mathematicians come up with stuff that’s crazy. Like string theory

Craziness credit should be given where it’s due: physicists came up with string theory. Surprisingly, they have used it to prove a bunch of interesting theorems in math but have (for the moment) mostly failed to do physics with it.

[C]an string theorists not propose any test whatsoever that would prove their hypotheses or is it rather the case that we lack the capacity to perform the tests that would figure it out?

I don’t think there is a scientific consensus on the answer to this question. But mathematicians doing mirror symmetry don’t care either way.

if we stipulate that humans invented the concepts 2, addition, equality, and 5, could they refine or alter their ‘invention’ in such a way that 2+2 did, in fact, equal 5?’ Obviously, totally, self-evidently not, right?

2+2=5, no. But 2+2=0, why not? That’s how computer count after all. Anecdotally, as the other thread was utterly ruined by my Barbie quote, I was doing math in an MRI in order to test whether math intuition was based on the neuronal recycling of (some parts of) the visual neocortex. That’s Stanislas Dehaene position. Or maybe the recycling is mediated by the language areas. That’s what Elisabeth Spelke thinks. In both cases, one can imagine fundamentally different cognitive properties (say, those of a bat, to take the usual example) leading to a radically different mathematical intuition than the human one. What’s math made of, ding-dong? How about the neuronal recycling of our ability to recognize faces?

In common with some earlier commenters, I don’t understand much – maybe anything, actually – of what the OP or many of the follow-up comments were talking about, sigh sigh sigh, and I actually have a doctorate in Classics. This makes me feel even worse, of course.

Is it a question of whether there is some [Platonic] Idea of numericity, etc., along the lines of the [Platonic] Idea of courage? It seems to me that numericity is embedded quite profoundly in reality (along with such things as proportionality, relationality, multiplicity, dearth) but that there have been various ways of apprehending/representing this feature since … well, I dunno – the Sumerians, perhaps? The capacity for high-level apprehension and eventually, symbolic representation of this numericity (or whatever) is, like the language faculty, [primarily] a human one – both language and mathematics depend heavily on man’s capacity for abstraction-generalization and symbolization. In this sense, mathematics, as man’s most abstract form of symbolic thought (cf. ‘pure mathematics’), it seems somehow to qualify as the ideal Platonic Idea.

Or not. But where would this get us, even if it were the case (as someone’s already noted)? We already know that mathematics is the most abstract form of thought man has yet developed …

Re: Cathy O’Neill, I’m a big fan – she’s a professional quant and former math professor (Barnard) whose chief interest these days seems to be Big Data and its dangers (in fact, I was expecting that when CT did a round of posts on Big Data, she’d be invited to guest post.) She’s a very strong advocate for women in mathematics, and travels around the country constantly, lecturing and encouraging women mathematicians. If I had known about what she calls fast / slow mathematical thinking when my own daughter was in school, I would have encouraged her a lot more to continue with math at university. She fell victim to testing both inside and outside the classroom; in despair, we hired a tutor, who finally figured out what she was doing for every single problem she encountered: she was deriving her own formula for solving it.

Finally, since somehow (but, how?) the subject of games has arisen on this thread, Slate has a nice piece up about the hardest game ever – only around 34 people have ever finished it. Had I known about it ten years ago, I’d have given both my children Robot Odyssey to play – together.

Like some other commenters, I don’t see why I have to choose between platonism and constructivism, even in math. I also don’t think one should encourage in impressionable minds (though I bet Zoë and Violet are anything but) an equation between “this is real; we didn’t make it up” on the one hand and philosophical commitments to realism/platonism on the other. Outside math, especially, this leads to madness, of both realist (modus ponens) and anti-realist (modus tollens) kinds. That we failed to get clear on this in the 90s is what makes the current “realist revival” (someone mentioned Meillassoux above) so painful (“ancestrality”? really? we didn’t beat this to death 20, 30, 40 years ago?).

dbk: This is only a fancy way of saying I believe numbers and mathematical operations can be said to exist in some meaningful sense independent of human intelligence, and when people figured out how to use the number [hey there’s nothing in here], i.e. zero, they were in some sense discovering zero and not inventing zero. Platonism about everything ever, and the Great Chain of Being, and the whole nine yards tends to be a bit much to swallow, but if there were ever any discussion you were going to get into with Socrates in which you were convinced rather than just bullied around by the world’s first proto-internet troll, it would be one about math. “And triangles, the fact that they all have three sides, isn’t this true about all of them? Could we make a triangle that did not have three sides?” “No, certainly that would be something else like a square” [good call Adiemantus!–ed] Etc.

I believe numbers and mathematical operations can be said to exist in some meaningful sense independent of human intelligence

that’s fine, but if you don’t say exactly what sense that is – which is *very* difficult – then you can unwittingly sow the seeds of some nasty and persistent confusion later on, of which the (cap-P Platonic) Great Chain of Being is only the most extreme (though interestingly not the worst) example. That’s all. Probably won’t come up, but you never know.

Since Zoe’s learning Japanese, I wonder, whether Japanese has the thing Korean does [1] where you don’t say a number, you have to add a syllable to say what kind of thing it is after the number (people, slips of paper, whatever)? Korean also has the issue w/r/t number words that there are two sets of number words and they’re used for different things even with the same syllable suffix, which Japanese most likely does not. It occurred to me to wonder whether that sets up different issues about abstracting numbers from concrete sets.

[1] In Korean I can say hello and thank you and “two tickets” and that’s about it.

It occurred to me to wonder whether that sets up different issues about abstracting numbers from concrete sets.

I don’t think it does, though, because that linguistic phenomenon is probably a syntactic reflex of the fact that numeral “adjectives” are not usual adjective at all, and that syntactic reflex is present in many languages, including English and Romance (think about rigid placement, overlap with determiners and absence of inflection in languages where adjectives are normally inflected, say tense inflection in Japanese or number and gender inflection in Romance).

In the image of the cave, in the Republic, it seems like the mathematical objects, including the pure numbers of thought, would have to be *within* the cave, among the shadow-casting models being paraded along behind the backs of the prisoners.

To think that numbers would lie outside the cave, in the realest of realms, is to fail to observe the distinction between ta mathematika and ta eide.

It occurs to me that I’ve only heard about platonism because of physicists and I’ve only heard of constructive mathematics because of computer scientists.

It’s not actually surprising that mathematicians themselves don’t particularly care about foundations–they’re content to play in their system of axioms. I don’t actually want to think about whether my appreciation of a work of art is based on any sort of objective value while I’m actually in the museum. But once you actually want to do something with math, some stuff becomes nice to know. If you define a theory of planetary motion using mathematics, do you have to worry about mathematics working differently on a different planet? If you want to implement mathematical theory in a computer proof assistant (which you would like to do because all of your physicist friends tell you that mathematics is required to understand the physical world and you want to make machines that solve real problems) then arguing over foundation is literally arguing over choice of programming language–and choice of programming language actually matters a great deal in an engineering project.

These are not non-problems. I only maintain a recreational interest in this stuff, but I think there’s some exciting stuff happening in constructive mathematics. A practical use for platonism is much harder to find, of course. But as I (naively) understand it, physicists are interested in string theory more because of it’s mathematical beauty than any falsifiable predictions made thus far. If string theory could later make such predictions, and the predictions held true, would that not be evidence for some kind of small-p platonism? If it is not merely the consistency, but the beauty of mathematics that makes it epistemologically useful, that suggests that the kind of structure we use mathematics to describe is somehow prior to physics.

The problem people have with whether or not math is true seems to be because we use the word truth to describe the conclusions of both deductive and inductive arguments. One type of truth is based on empirical experience, e.g. It is true that the speed of light is 299,792,458 m/s. Another type is based on if a statement conforms to a set of rules, e.g. It is true that 2+2 = 4. The trouble with platonism comes from confusing the first type of truth with the second. Empirically, existence is equivalent to truth, but it does not follow that logical truth is equivalent to existence.

As to 2+2 = 5, there is a thing called paraconsistent logic, which is “inconsistency-tolerant,” 2+2 = 5 could be taken as true. And there are interesting conclusions you can derive from these premises.

bianca @99. Thats interesting. So it sounds to me -being at heart a physicist, that as described the Korean language doesn’t support the concept of a dimensionless number. For those with maybe an education in the classics, a dimenional number is something like c: the speed of light, and it has values of velocity [length/time], whereas anything that is a ratio of numbers of the same dimensionality is dimensionless. So pi is dimensionaless -it can’t be expressed in terms of lengths time mass. So how well do Koreans do with physics, given that they started out with a language that didn’t allow the concept of dimensionless?

Instead of asking if numbers exist we should define existence in a way that is mutually agreeable and then define numbers in a way that is mutually agreeable and then ask if the the former is a condition of intelligibility for the latter. Or something.

From the Hamming piece (which I didn’t on the whole find very insightful but I like this bit):

“Just as there are odors that dogs can smell and we cannot, as well as sounds that dogs can hear and we cannot, so too there are wavelengths of light we cannot see and flavors we cannot taste. Why then, given our brains wired the way they are, does the remark “Perhaps there are thoughts we cannot think,” surprise you? Evolution, so far, may possibly have blocked us from being able to think in some directions; there could be unthinkable thoughts.”

(Perhaps one could make up a number-suffix that meant `dimensionless’?)

For example for a long time in the mathematical practice that grew out of that of the Hellenistic world, “numbers” as in “five flower petals” where considered distinct from “numbers” as in “the length of this side of a triangle is five” – there was no concept of ‘number’ as we use now that can be used to either count or measure.

This perplexes me, as I believe I could (in spring) measure by laying flower-petals along my line and counting them. Yesno?

As to Belle’s question….if I interpret “+” as “add the next number plus one” then “2+2=5″. Easy, done. Conceivable. I just conceived it!

This doesn’t show what you seem to think it shows. Obviously, we refer to mathematical entities and operations using certain symbols, and you or I or anyone else could decree that they’re going to take some such symbol and use it to mean something else—to assign it a new reference, essentially. That by itself tells you nothing about the mind-dependence of the operations or entities in question. Obviously, mathematical cognition, if such truly exists, depends on am ability to use abstract symbol systems, but this by itself couldn’t possibly settle the question with regard to Platonism, because it tells us nothing about what the proper semantics are for the relevant symbol systems.

Also, it doesn’t actually show that ‘2 + 2 = 5′ is conceivable. Any more than I can make true that the Moon is made of green cheese by decreeing that I’ll use ‘the Moon’ to refer fried balls of green cheese, let us say.

I really dislike that quote.Thoughts are very different than pure senses like smell or vision. There are physiological reasons that we cannot see certain wavelengths. Given that thoughts are constructed out of an arbitrary system of representative signs, what could conceivably physiologically constrain what those signs could represent?

I’ve never understood how Max Tegmark can propose the existence of universes of platonic entities with no evidence whatsoever, and have his work published in Scientific American, while Rupert Sheldrake’s theory of morphic resonance, which at least has some evidence for it, is ridiculed by the scientific establishment.

I think I haven’t explained myself well. I’m not claiming that the mind-dependence of the operations in question is settled by my silly example, but I am trying to point out that the question of truth is kinda moot, since the answer is already baked into the question. That is, my silly answer is totally unsatisfactory because you already have a strong sense of what the symbols must mean. And that meaning pretty much includes taking 2+2=4 as part of the package. You are declaring that we have to stick to the rules of the game at the start, and then asking whether we have to stick to them.

A technical question: does mathematics have a semantics, or just a syntax? (I don’t know the answer, but am just wondering what those who do mathematical logic or meta-math would say).

At any rate, Belle’s worry seems to me to be the sort of thing that Rorty would respond to with, “if it don’t itch, don’t scratch”. I could be that one could make sense of “2+2=5″, via ambiguous re-definition. (It was a more sophisticated version of such that got this round of the business going). Adding logs is the same as multiplying integers, so some similar expression could occur, if one could somehow “count” with logs or define what the “unit” would be. But the worry is the same for language. Perhaps someone might be tempted to speak of “dry rain”. (What would that be? Sleet?) But the rules constituting such expressions make that exceedingly unlikely, as they must ultimately be in some sense useful. But what is useful doesn’t eo ipso translate into “objective existence”. It seems to me that the worry results from that old distinction between the natural and the conventional. And the failure to appreciate how deeply the “conventional” penetrates into the “natural”, (and how bootless, in this age of pervasive technological manipulation, appeals to the “natural” are). As well as, a basic fear of contingency. The worry simply doesn’t occur at the level stipulated and is misplaced. And, on the other hand, it lends itself to such enormous reifications such as, “the universe is a giant computer program” or “the economy is a giant calculating machine”, that are precisely to be resisted as the basis for understanding anything, that reduce understanding to technical manipulation.

Also the question as to whether symbolic “realities” are “mind-independent” or not seems misconceived. Not only does it imply or presuppose some peculiar conception of “mind” and its own independent status, but symbols are precisely “things” that are exchanged communicatively between organisms across the world, and thus are also involved in materially embedded practices that themselves are not necessarily conscious or intentional, (while rendering at least partly the latter possible). Factoring out that mediating “praxiological” realm is what makes for a mistaken binary alternative.

Also it’s curious that the example for an objectively existent function that Bell chooses is “0”, nada. (Though I myself have wondered how they did trade and debt accounting without it, how they registered losses as well as gains. Did they simply just assume that stealing more booty and capturing more slaves would always allow them to “make good”?)

Probably that pattern recognition, a comparison of more or less, than formally speaking, counting. Monkeys can “count” up to 7. Coincidentally, phone numbers in the U.S. were restricted to 7 digits because experimental psychologists determined that that was the maximum that could be readily remembered.

To change the point of view a little bit, what if we the sum of the angles of a triangle was not 180 degrees?

At first glance this might seem as absured as asking what if 2+2=5, but once you take it to heart it leads to something useful (and arguably more physically relevant) — non-euclidean geometry. Perhaps there’s some deeper meaning to this?

You are declaring that we have to stick to the rules of the game at the start, and then asking whether we have to stick to them.

I see your point. And frankly, I don’t have any knock-down argument against constructivism. But I do think I wasn’t really asking whether we have to stick to the rules—rather pointing out that the rules in question seem inescapable. And even there, I wasn’t so much forwarding an argument as articulating what seems a rather strong intuition for a lot of people, one that I myself am inclined to account for in normative vs. naturalistic/psychologistic terms.

I think I’m mostly inclined to agree with what you say. If I’m understanding you correctly. I especially agree that the ‘natural’ vs. ‘conventional’ contrast is fairly well hopeless. And at one level of explanation, I’d even agree with the it’s-all-just-formal-rules-and-their-implications You are declaring that we have to stick to the rules of the game at the start, and then asking whether we have to stick to them. (Sort of.) I suppose my thought is something like: these rules are reflective of certain very basic structures of thought/reasoning and so are not ‘constructed’ in any but the most tenuous sense. I’m very sympathetic to a parallel thesis about the grammar/syntax.

As for whether mathematical statements have a semantics—I suppose this is contested? It seems to me natural to grant that they have semantic purport in some sense, given how intuitively we can make sense of preserving meaning across translation, talking ‘about’ the same thing while using different symbols, etc., but of course this is not incontroversial.

I think so. I was thinking of folding a sheet of paper, or perhaps a piece of string. And in Peter T’s link about birds it appears that cormorants can count up to 7, presumably without numbers!

Also, @ 110

The Hamming talk includes fascinating example of deductive reasoning vis-a-vis Galileo and the rate of falling bodies. It also includes this gem, (paraphrasing) “once we move from integers to fractions we have to abandon the idea of the next number.”

126: He said “signs” not “words” right there in the text. Yup there it is “signs”

Wiki

In semiotics, a sign is something that can be interpreted as having a meaning, which is something other than itself, and which is therefore able to communicate information to the one interpreting or decoding the sign. Signs can work through any of the senses, visual, auditory, tactile, olfactory or taste, and their meaning can be intentional such as a word uttered with a specific meaning, or unintentional such as a symptom being a sign of a particular medical condition.

peterv 01.27.14 at 11:16 pm @ 115 — Tegmark claims he has some sort of evidence. (I haven’t read it.) Also, in the material I’ve read, his views are presented more as hypotheses than as a well-established theory derived from evidence and logic. I was unaware that there was any evidence for morphic resonance; I have read of experiments which were supposed to demonstrate it and failed to do so. I got the idea it was another Secret Life Of Plants sort of thing.

No, Frege’s theory of integers and the like. Establish the interval between 0 and 1, then loop that recursively through to 2 und so weiter. Or something like that. It’s describable as a set of formal operations and infinitely repeatable. Same or different is not equivalent to”logically identical” or equal. It’s an analog process, not a digital one, and likely draws on different mental/neural bases.

Demanding that I make some sort of reasonable definition of “exists independently” would both be completely fair and cause my enterprise to fall to pieces faster than the pins of regular Platonism with Aristotle tossing a bowling ball of that pesky third term down the alley. That’s why I di’nt do it. I’m not suggesting anything like anything that’s even gesturing vaguely towards a proof; this is like a feeling. More Than a Feeling. O HAI ITS BENZENE is not meant to prove there are numbers, just that structures exist independently of humans. On which grounds it succeeds. Also thanks for thinking it’s funny. Zoë and Violet say already that in Mandarin there are different counting words for different kinds of words. Not different numbers, clearly, though. Just different classifier words that go with the number and the noun because there’s no plural forms because–hey!–there’s no forms! Easy-peasy! Sorry about that whole, um, writing system, kids.

And the tones. Although, they say, only four is no big deal and you learn them right away but they think Cantonese sounds scary and when I made native Taiwanese dialect’s “creaky” tone they fell out laughing. I’ve decided to learn Japanese with Zoë. No pesky tones, a moderate degree of alphabet-like items (katakana + hiragana is 100 or so?), kanji…eh I’m good at memorizing symbols, hideous complicated verb system: muahahahaha….steepled fingers of triumph…now we enter my terrain. I. Will. Prevail! Also, I’ll read lots of Sakura Hime and thus have to learn the verb forms you only use when you’re asking the Togu, in the politest and most remote conceivable terms, if he might not condescend to consider doing the thing you merely suggest as one of a palette of many options. (The Togu is the Crown Prince, but for real he’s no question going to be the Emperor, this shit is REAL PEOPLE DON’T PLAY it’s the TOGU.)

I think the point is that magnitude/multitude, continuous/discrete, intensive/extensive are/were useful distinctions. And maybe some would argue that since Descartes they have sometimes become blurred.

Apologies for dumbing down the discussion, but does anybody know of a language that does not have dimensionless numbers? (I am going out on a limb to say that I highly disbelieve that Korean doesn’t have them; Japanese certainly does.) Even the most literal-minded individuals seem able to make use of them, which is interesting.

This is probably of no use for the constructivism / platonism discussion above, of course.

With regard to paraconsistent logics, it seems to me that if you can get “interesting” results, the subset of “useful” results you get are actually obtainable in a more rigorous system – the paraconsistent logic just being useful (I would assume) as a way to short-circuit some of the usual processes (but why?). Also wanted to throw a shout out there for Armando mentioning Godel’s hurdles.

You know, Belle. So many people have come and gone, their faces fade as the years go by.

Which brings me to my theory. The appeal of platonism (never forget, btw, Plato was the O-riginal Republican) is that people really, really have a problem with the fact that everything in this world is only here for a moment…

I studied logic with Chris Mortensen. He wrote a book, Inconsistent mathematics (Kluwer, 1995). It’s been several years, but as I recall, he demonstrated that you could build an analog to a Robinson arithmetic on a paraconsistent logical foundation. I think the idea is to get away from the inconvenient paradoxes you get with classical logics. As for its “useful” results, Mortensen said you could build a paraconsistent mathematics good enough to kill people with, so there’s that.

Just a note which is sort of obvious, but I thought I’d made explicit – I’m pretty much a formalist. The way to mock that position is to say that I believe mathematics is largely meaningless squiggles on a piece of paper/tablet/blackboard. But I really think there is something to the notion that you separate out your sums from the things those sums are modelling.

This is what kids watch now? Well… at least it’s not the Hasbro GI Joe/Transformers Power Hour.

Let me take a less constructivist/instrumentalist/Platonist standpoint and answer the main question – “are numbers real, or are they just something people made up?” – all Zen-like with another question: Is race real? Is capitalism? Beauty? Injustice? Is green real? I can make pretty solid arguments that those things exist only because of human action and human categorization. Would the world make more sense if you denied those things existed? I think this is way better than doing Frege and Peano because when you go down that route, you end up finding Gödel and concluding that no arithmetic can be provably consistent anyway.

Besides, I wrote an essay once that recast the whole history of the foundations of mathematics in Marxist terms as a succession of class revolutions, moving from feudal-monarchical Platonism to industrial-imperial age Fregean logics, on to Bourbaki’s post-WWI “New Math” for the proletarian “working mathematician”, stagnating for years until a new generation of leadership ushered in a period of glasnost and perestroika after computers took off in the 70s and 80s, and finally leading to modern anti-foundationalism and the 21st century altermondialist computational-algorithmic neofoundationalism of guys like Chaitlin and, in a pinch, Wolfram. I think my version of that history makes at least as much sense as any mathematicians’ treatment, so it’d be hypocritical of me to start rejecting a humanities approach to the problems of mathematics now.

You are declaring that we have to stick to the rules of the game at the start, and then asking whether we have to stick to them.

If mathematics isn’t real, then “have to stick to the rules” has no meaning. We agree upon a set of rules. I say the rules lead to this conclusion. You say they do not. Is there a fact of the matter as to who’s right or wrong?

@ 143 — I’d like to read the essay. Otherwise, the question, ‘Whaddaya mean, ‘real’?’ has been raised and the response(s) indeed seem(s) like (a) first step(s) towards clarifying the formalist-realist-whatever debate. (Inflections can be annoying, sometimes.)

Just so I understand some of the stronger anti-platonist positions here, is anyone claiming that mathematics is arbitrary, but the rest of science is still perfectly real and useful? So that we might encounter aliens that have the same theories of physics and chemistry that we do, but it is absolutely impossible to translate between their quantitative system and ours?

Anarcissie@145, I have a chance at a job that would maybe let me wrestle it into publishable shape and I really want to because I think it has the spark to be a very good, if somewhat farcical, essay. If I get this job, it goes into the packet I send to anyone who wants to be my editor.

Consumatopia@146: I don’t actually think that, but most people think I’m an anti-realist about science anyway.

But let me make a Lakotos-Putnam style argument for the sake of devil’s advocacy: Mathematics is quasi-empirical because of its connection to science and technology. Science and technology are real because, well, airplanes and stuff, but seen on its own, math is pure mind games with no reality of any kind. It only takes on any kind of truth when bound up with some application. Even 2 + 2 = 4 is only true because I need to know that my brother isn’t getting more M&M’s from the packet than I am.

If mathematics isn’t real, then “have to stick to the rules” has no meaning.

Negative, Captain. Also, how are you using the word “real” here?

Just so I understand some of the stronger anti-platonist positions here, is anyone claiming that mathematics is arbitrary, but the rest of science is still perfectly real and useful?

Mathematics is extremely useful. I don’t think anyone is denying that. The anti-platonists are saying just because it is useful doesn’t mean it has any existential status comparable to matter or energy. It is a way of thinking. Lots of ways of thinking are extremely useful!

So that we might encounter aliens that have the same theories of physics and chemistry that we do, but it is absolutely impossible to translate between their quantitative system and ours?

We don’t need aliens to follow this example to its logical conclusion. Consider that birds must have some notion of fluid dynamics to fly, or that termites must have some notion of gravity to build mounds. Surely their physical notions must be compatible with ours. They can’t have a fundamentally different science that would let them do magic. But do you expect them to have scientific theories that look anything like that ones I learned in university?

Must they? A lot’s going to depend on what you mean by “notion” here of course—they must have the ability to stay afloat, navigate, etc., and you could take it “having some notion of fluid dynamics” simply follows from that. But I don’t think that affects Consumatopia’s point at all, which (I think) requires some at least minimal theoretical grasp of what makes abilities like staying afloat, etc., possible. And I don’t see any reason to think that birds have a “notion of fluid dynamics” in this even minimally theoretical sense.

Oh, if that’s all, then I’m totally on board. But I think that’s not so much Platonism but Hegelianism: “The real is the rational and the rational is the real.” The rational is the order or organization that can be discovered, pointed out, or described, whether by math or language or picture making.

In a way it’s anti-Platonism, since in Plato structures exist not just independently of humans but of the material world, whereas I assume you mean they exist independently in the material world.

(I actually read Hegel this way too, since I think he’s a dynamic monist, not a dialectical dualist–the opposition of form and matter is an illusion of becoming.)

… requires some at least minimal theoretical grasp of what makes abilities like staying afloat

People had aerodynamics before they understood why atoms exclude each other from occupying the same space and I’m not sure the existing answer to that question is anything more than a complex kind of question begging. (Why do atoms exclude each other from occupying the same space? Because they are fermions. Why do fermions exclude each other from the same space? Because they obey the rule that no two particles can have the same quantum state. So why can’t two fermions have the same quantum state? Because if they could do that, they wouldn’t be fermions.) I could imagine an alien saying humans have no understanding of why their airplanes work because we don’t have a complete theory of the universe.

I do think that birds know when they are flying and when they aren’t. I think they have some clear ideas about what air is and how flapping their wings makes them fly. They must have a model of the world and their model of air and motion must be consistent with ours, at least over the range of phenomena a bird is likely to encounter. I’m not sure sure you can define “minimal theoretical grasp” in a way that doesn’t imply “did a bunch of math”, in which case you’ve defined the knowledge it takes to do science as including math as people usually understand it and the claim becomes tautological.

They must have a model of the world and their model of air and motion must be consistent with ours, at least over the range of phenomena a bird is likely to encounter.

This actually strikes me as rather implausible, but (a) I’m not sure that I have a good argument to back this up, and (b) I imagine I’m very much on the minority on this anyway. But yes, if you think that birds must have a model of the world with component models of air, motion, etc., then what you say in 149 makes a lot of sense.

Mathematics would be real if when two people disagree as to whether following a set of mathematical axioms leads to a given conclusion, at least one of them is objectively wrong. Grant me just that alone, and the work of mathematicians becomes meaningful, and the application of mathematics to the sciences is justified.

(Mathematics would be more real if there were objective reasons to prefer some axioms to others, and even more real that that if those reasons were capital-N Necessary in some sense (“mind of God”). But I don’t you need either of those to say that mathematics is as real as matter/energy.)

The anti-platonists are saying just because it is useful doesn’t mean it has any existential status comparable to matter or energy.

Math is obviously not a physical object. But if it doesn’t have existential status that’s at least comparable to gravity, then it must be dispensable–it should be possible to describe the laws of physics just as well without math (or anything isomorphic to math). Otherwise, it’s a truth of the physical world that something like math is needed to describe it.

I do think that birds know when they are flying and when they aren’t. I think they have some clear ideas about what air is and how flapping their wings makes them fly. They must have a model of the world and their model of air and motion must be consistent with ours

I’m with js. It looks to me like you are adding unnecessary components. I don’t need any abstract models to become highly proficient at throwing a baseball or running through the jungle. Models might help in some cases, in other cases they may be a hindrance. But they certainly are not necessary to execute physical operations with the body.

Anon 01.28.14 at 4:57 pm @ 152:‘… In a way it’s anti-Platonism, since in Plato structures exist not just independently of humans but of the material world, whereas I assume you mean they exist independently in the material world. …’

I think it’s a misfortune that mathematical realism has become conflated with Plato and his works, since the latter come with a lot of (to me) unpleasant baggage. The notion that some sort of higher realm of forms both exists completely independently of the material world and at the same time produces it, ‘a spume that plays upon a ghostly paradigm of things’, is clearly self-contradictory: if A produces B, A is not independent of (disconnected from) B. Just so, the sense that when I do mathematics I am discovering rather than constructing or inventing something, leads me to believe not that the mathematics is separate from the material world, but that they are two aspects of one thing, which become separately visible because of the way they are looked at.

Mathematics would be real if when two people disagree as to whether following a set of mathematical axioms leads to a given conclusion, at least one of them is objectively wrong.

“Objectively” is the troublesome word here. Since math is a collection of rules, a wrong answer in math is “conventionally” wrong. At least I think that is a better approximation.

Math is obviously not a physical object. But if it doesn’t have existential status that’s at least comparable to gravity, then it must be dispensable

Well, clearly, it is dispensable in the sense that we could survive without it. Similarly, the world doesn’t need to be described. Plants and animals can thrive without telling stories about the world. Math does an excellent job of describing the world. No coincidence since that is what we devised it for. But we don’t really know how accurate our mathematical descriptions are. If history is any indication we can bet that what is considered mathematically accurate today will someday be discarded as “crude.”

“Mathematics would be real if when two people disagree as to whether following a set of mathematical axioms leads to a given conclusion, at least one of them is objectively wrong.”

Thats a very odd definition of real, there. In particular, the very silly example of addition I gave above would certainly pass this test for real-ness, without much effort. It also allows for contradictory things to both be real, which is fine, I guess, but somehow removes the point of a lot of it.

One of the reasons Platonism isn’t as popular as it used to be is the advent of non-Euclidean geometry. Somehow, a modern mathematician needs to accept that both Euclidean and non-Euclidean geometry are equally valid. This is actually ok, but since the Platonists put quite a bit of effort into saying that Euclidean geometry was obviously the real one, and non-Euclidean geometry was obviously false, since it didn’t reflect reality, the Platonists position has suffered somewhat.

For what its worth, this definition of real more closely looks like logical consistency, which is mostly associated with formalists (who don’t think mathematics is “real”). So this path looks kinda backwards, from the point of view of mathematical philosophy.

I wouldn’t say that mathematics, of itself, describes anything at all. The conventions come into it, when we want to use math to describe the world. We need conventions to make that possible. Those conventions have to do with making a social agreement on the correspondences to the observable world, which make it possible to communicate a description. The conventions used to describe mathematical concepts and reasoning are of this same kind, social agreements on a correspondence with what is commonly observable. It’s Eve pointing out things to Adam and slapping him alongside the head when he’s obdurate about naming names. But, math isn’t the conventions any more than a rabbit is the name of its kind, “rabbit”.

@160, describe how your definition of “+” applies to everything else it applies to, and then of course it’s real–you could make a computer program that performs it. The computer program is operating on a real, physical machine, and therefore by describing the behavior of your “+” operator, I would also be describing the behavior of that physical object. How much more real can you get?

There’s the further question of “which + is really addition?” But you can paraphrase this as “is there any reason to prefer the conventional ‘+’ operator to Armando’s re-definition?” And that question of utility is a real, empirical question–you could force a group of people to use your strange ‘+’, and see what they manage to accomplish with it.

“Objectively” is the troublesome word here. Since math is a collection of rules, a wrong answer in math is “conventionally” wrong. At least I think that is a better approximation.

No, objectively is definitely the right word (and if you insist on a definition, I’ll pick “not conventionally”.) Math is not a particular collection of rules–see Gödel. You might, as a matter of convention, select a particular group of rules. But it is not the case that because the set of rules you pick is conventional, the truth of whether or not you are following the rules must also be conventional.

But we don’t really know how accurate our mathematical descriptions are. If history is any indication we can bet that what is considered mathematically accurate today will someday be discarded as “crude.”

If the math is somehow “inaccurate”, then our descriptions of the world must also be inaccurate.

Look, it may be the case that all our scientific theories are wrong. It may be the case that there’s some way to replace all of the math in all of our theories with something else that does just as well without being equivalent to math. But those are both empirical questions, and if neither of them is the case, then math is real.

@126 “Are you saying that we can’t think any thoughts for which there are no words? If so, is it an uncontroversial assertion among people who study this stuff? I wouldn’t have thought so.”

Well, there are two ways that people talk about mental behavior. Sometimes by a thought they mean a unit of subjective experience and sometimes they mean a unit of private linguistic behavior. So if you mean the first definition, then you can have thoughts for which there are no words, but if it is the second you can’t.

Since mathematics is a linguistic phenomena, mathematical thoughts must be of the second type, so one cannot think about math without using words. People mix up these two definitions all the time, leading to all kinds of confusions.

But what about the babies and the crows that show numerical behavior but are not able to use language? I think the scientists doing these studies are sloppy with their language, and they should not be saying that babies or crows are “doing math.” I am more of a behaviorist than is fashionable these days, but simple numerical behavior is not at all the same thing as an adult human using language to study math. In the same way, any non-linguistic notion of aerodynamics that birds have is a completely different type of thing than the linguistic subject of aerodynamics that humans use.

If we met aliens and they used language, then we would be able to understand their math, after we succeeded in translating their language. (This basically leads us to Blindsight and thats a whole other discussion.)

In very recent times, you have the question whether it’s necessary for the even negative powers to be closed with respect to negative numbers, or whether it’s necessary for cardinality to be closed with respect to infinite sets. The topics raise existential questions. How to answer those questions raises more questions. (At this point, I think, it is actually possible for one mathematician to say to another that he seems to be following the rules but what he’s talking about doesn’t exist and is thus out of bounds.) Platonism itself appears to be more general.

I suppose Plato had beliefs about math, and those beliefs could be called Platonism. Present-day mathematicians probably don’t read Plato, so it would be surprising if they held the same beliefs as Plato.

There is a kind of almost-mysticism that goes along the lines, there isn’t any inherent reason to believe in i, yet when we believe in i, and take some further steps, we get something that matches reality, therefore i represents something in the fabric of the universe–which isn’t yet, or maybe can’t be, explained. I don’t know whether I’d call this “Platonism.”

The idea that math is made of Ding Dongs also raises existential questions, due to the bankruptcy of the Hostess Corporation. Even watching the video didn’t totally erase that interpretation from my mind.

Bruce Wilder @ 161: I wouldn’t say that mathematics, of itself, describes anything at all. The conventions come into it, when we want to use math to describe the world. We need conventions to make that possible.

5+5=10 doesn’t describe anything, the way “I have five fingers on my left and five fingers on my right and therefore ten fingers altogether” does. But (a+b)+c=a+(b+c) does describe the fact that when adding pebbles to a pile, the order they’re added in doesn’t matter. You’re the first one in this thread who’s interpreted “math” in that way. I believe everyone else who’s referenced “conventions” is referring to things like the use of base 10 numbers rather than base 16 and the use of “+” to mean “plus” rather than “logical OR.”

I’m not sure why you need a computer here. I can do things on a piece of paper, so in that sense they are real. This is an extremely low bar for reality, by the way. That is, if I can conceive it and write it down, then it is real. (I’d note that you wouldn’t apply that standard to fictional constructs. That is, you wouldn’t say that Narnia was real, just because you can write it down describe it etc.)

My silly plus can easily be written into a set of rules – mathematicians do this sort of thing all the time. It is really well understood how you can do this arbitrarily. It is also quite well understood that most mathematics you get by writing down random rules doesn’t really go anywhere, but thats by the by.

But sure, if you are saying that any set of rules you write down is real, because you have written them down (despite the fact that different rules are contradictory) then fine. This is about a million light years away from Platonism though, and is really the same position as the anti-realists. Essentially, if you are describing everything as real (I’m not sure you get how broad this definition really is) you might as well describe nothing as real. It doesn’t make any difference.

“A raven whose brain processes ‘I need friends to help me drive off these three interlopers. I have found one; I need more’ is dealing with a numerical reality, not an articulated symbol system.”

But in this case “three interlopers” isn’t a reality in the sense of a mind independent fact. The total number of interlopers in the universe is a mind independent fact, while the raven’s “three” exists only in relation to the raven. “Three interlopers” means three of them *here*, in relation to this raven.

This is why it’s also extraordinarily strange to say we “discover” zero or that zero is “real.” I hate to side with Sartre on anything, but consciousness brings nothingness into the world; something is only absent in relation to the intention of a consciousness that seeks it. There is no nothing.

I don’t recall Sartre’s specific example, but a true statement such as “Jean-Paul is not at the cafe” is not a mind-independent reality. Jean-Paul’s not-being-there is true only in relation to a “here” of consciousness. The mind-independent reality of this “not” is his being at the next cafe down the street. The reality of “I have zero apples” is that there are apples. There are no zero apples.

I’m not really a trained math person so I came into this question and have tried to follow the thread with an open mind. My initial reaction was that Belle was correct and numbers were “real” since, as per her example a Sakura would have 5 petals.

After thinking about it, however, I think numbers/mathmatics are more a language and more specifically a language of adjectives. They enable us to describe things and relate things to one another in a particular way.

If there was only one Sakura in the world, it would have 5 petals but the number of petals wouldn’t be a real thing on its own, the Sakura would be the real thing. Its only in relating the Sakura to other things that we look to its color, shape, mass, length and number of petals etc. All of these characteristics could be described using numbers – but they are just adjectives.

It might be weak support, but its interesting that people fluent in languages and adept at music are often also adept at math. If all are different flavors of language this might make biological sense as well. [I’m not a biologist if this last doesn’t make sense ]

But if it doesn’t have existential status that’s at least comparable to gravity, then it must be dispensable–it should be possible to describe the laws of physics just as well without math (or anything isomorphic to math). Otherwise, it’s a truth of the physical world that something like math is needed to describe it.

But that wouldn’t give mathematics any special ontological privileges. I think of it this way: the world behaves in a highly regular fashion. If you begin with that conception, then the first thing you do when you really start trying to understand how it works is to figure out what the regularities are. But the language of mathematics is perfect for describing regularities; you could write out something like the concept of a derivative in natural language if you like, but once you formalize it, you really have a nice regular framework that’s widely applicable. Then it turns out that lots of things in the world behave in such a way that they can be said to obey derivative-like relationships so naturally you take that function or structure or what have you and start using it wherever it looks useful. It seems to me that any coherent description of regularity would in some sense be isomorphic to a mathematical language, if only because you’re really concerned with creating structures that behave in the same consistent fashion that (you imagine) the world behaves, and mathematics is an excellent tool for that.

Armando @ 172: I would indeed say Narnia is real. As for the idea that fictional characters are real, I like the work of Amie L. Thomasson and her artifactual theory of fictional characters, which essentially argues that fictional characters are real and that they continue to exist as long as the texts in which they appear continue to exist. In other words, the continuing existence of the character is contingent on the text not disappearing. Work flowing from Meinong’s Jungle is also worth considering in the realm of whether fictional characters are real. Link to a book by Thomasson here http://books.google.com/books/about/Fiction_and_Metaphysics.html?id=HrM2ZsuCdG4C

I’m not sure why you need a computer here. I can do things on a piece of paper, so in that sense they are real. This is an extremely low bar for reality, by the way. That is, if I can conceive it and write it down, then it is real. (I’d note that you wouldn’t apply that standard to fictional constructs. That is, you wouldn’t say that Narnia was real, just because you can write it down describe it etc.)

The computer has one big difference compared you with your piece of paper–it’s behavior can be simply predicted. What matters is not the symbols, but the behavior of the symbols. I can write down a definition of a super-turing oracle, but I can’t predict what it would do, because it’s not real. (If I could predict it, then I would by definition have a super-turing oracle, and therefore it would be real.)

My silly plus can easily be written into a set of rules – mathematicians do this sort of thing all the time. It is really well understood how you can do this arbitrarily. It is also quite well understood that most mathematics you get by writing down random rules doesn’t really go anywhere, but thats by the by.

It’s not by the by, it’s the entire point–any set of rules can be arbitrarily selected and written down, therefore the real business of mathematics is not arbitrarily selecting a pair of rules and writing them down. If mathematics were just a matter of convention (the anti-realist position), you would go just as far with any set of rules.

But sure, if you are saying that any set of rules you write down is real, because you have written them down (despite the fact that different rules are contradictory) then fine.

Any set of rules that you can completely describe is a real rule. That a rule implies a contradiction is a real fact. The implications of the rule are not necessarily true, but that the rule implied them is.

This is about a million light years away from Platonism though, and is really the same position as the anti-realists.

Formalism is much closer to what most people who call themselves mathematical platonists would call platonism than it is to, for example, the Lakoff/Núñez claims about mathematics as a metaphor. A formalist doesn’t care whether babies “naturally think logarithmically”–it’s just a matter of sticking to whatever the defined rules are. The anti-realist would deny that there’s any sense in “sticking to the rules” other than what the neurons in our human brains do when stimulated by the image of the rules on on paper.

Then it turns out that lots of things in the world behave in such a way that they can be said to obey derivative-like relationships so naturally you take that function or structure or what have you and start using it wherever it looks useful.

I guess if I say “derivatives are real”, that’s equivalent to “lots of things in the world behave in such a way that they can be said to obey derivative-like relationships”. It doesn’t matter if those derivatives are described by natural language or funny symbols.

I thought the original claim wasn’t that the ancient Greek mathematicians could distinguish magnitude/multitude, etc., but that they were incapable of translating between them. And this is odd, because straightedge-and-compass constructions are useful for teaching because many people can naturally interpret them in either sense. I agree that analytic geometry blurs them even more, but distance in the number of swings a compass makes seems as natural as distance in the number of steps my legs make.

Cephalopods seem, IIRC, to *not* naturally have this — they have superb haptic sense with scant or no sense of distance. I don’t expect interstellar aliens, but I would be so happy to know what cephalopod math could be. They solve what we think of as puzzles, so they might even have models of the world at some level. When I read

They must have a model of the world and their model of air and motion must be consistent with ours, at least over the range of phenomena a bird is likely to encounter.

I agree with it, but I assume their `model’ is not like an aerospace engineer’s; some of it might even be built into their senses. …Someone must be spoofing some but not all bird-senses to see how they respond. Hm. Nevermind. Anyway, lots of *humans* are very efficacious at doing complicated, learned-expertise problems but can’t explain their (presumed) model, or even insist there isn’t one — I know nurses and doctors who report their problem-solving as like another sense, not like a chain of reasoning. Usually I can pull a chain of reasoning out of them, but neither of us knows how much of it was retrofitted.

Then when A H writes

Thoughts are very different than pure senses like smell or vision.

… how do we know? Could we ever have had a report of thought from an entity that hadn’t been primed with sensory input? ….OH, as I was writing A H has written more: perhaps [A H]-thoughts are linguistic thoughts? That doesn’t seem sufficient to describe doing even math or programming, even though I can retrofit linguistic descriptions onto my non-linguistic intuitions. I should reread William Calvin on movement and language and brain activity. (And music, I think, Trader Joe.)

I guess if I say “derivatives are real”, that’s equivalent to “lots of things in the world behave in such a way that they can be said to obey derivative-like relationships”. It doesn’t matter if those derivatives are described by natural language or funny symbols.

That seems like a very pragmatic outlook to me, and one that I would agree with, but I don’t think it’s what people have in mind when they talk about mathematical Platonism. Those claims are usually much stronger, e.g. “Derivative [or numbers or whatever] are actual ontological features of the world,” not merely “useful descriptive formalisms.”

@179, It may not be full Platonism, but it makes two claims that seem to be in dispute: that the term “derivative” refers to a single thing whether it’s described in symbols, language, or some alien communication system; and that in this world there are human-independent facts related to derivatives just as there are facts related to matter and energy.

@179, It may not be full Platonism, but it makes two claims that seem to be in dispute: that the term “derivative” refers to a single thing whether it’s described in symbols, language, or some alien communication system; and that in this world there are human-independent facts related to derivatives just as there are facts related to matter and energy.

With regard to human-independent facts, the way I would describe my position is that there are processes that are adequately modeled by some corresponding mathematical formalism. There may exist “facts” about derivatives, but they exist only within the context of a particular mathematical framework; they aren’t facts about the world as such, just facts about certain theories. Meta-facts, perhaps?

@178 “Could we ever have had a report of thought from an entity that hadn’t been primed with sensory input? ….OH, as I was writing A H has written more: perhaps [A H]-thoughts are linguistic thoughts? That doesn’t seem sufficient to describe doing even math or programming, even though I can retrofit linguistic descriptions onto my non-linguistic intuitions. I should reread William Calvin on movement and language and brain activity. (And music, I think, Trader Joe.)”

A major problem with discussing this is that our neuroscience is not very advanced. We just don’t know what the neural basis of subjective consciousness, willful action, or language is. So it is hard to say exactly what the difference is between sensory experience and linguistics acts, because at a fundamental level we don’t even know where these mental phenomena come from.

However, I think a lot of the problems come from basic category errors like I described above. Linguistic behavior is built out of non-linguistic mental acts. Any sentence that is spoken or thought is produced from a non-linguistic source. It can’t be true that we verbally plan out every thing we say, or else there would be an infinite regress of planning sentences. Like I said above, it is a genuine mystery how this happens, though I am optimistic it will be solved within the next 50 years or so.

So doing programing or mathematics is not a purely linguistic, however the content produced by these activities is purely linguistic (music is an interesting edge case). The problem with the Hamming quote was that she grouped limits on non-linguistic mental acts, hearing and smelling, with limits on the content of linguistic acts. So maybe it is true that if you think by speaking words in your head, you could not think in sounds beyond your range of hearing, however this does not mean that the content of your thoughts is an anyway limited. The fact that language is made up of arbitrary signs, means that there are no limits what it can represent, because it’s arbitrariness means anything can be given any sign.

A H 114: “Given that thoughts are constructed out of an arbitrary system of representative signs, what could conceivably physiologically constrain what those signs could represent?” We don’t have a good account of how thoughts are represented in nervous systems – wait we don’t even have a good account of what thoughts ARE – so how can we know how limited we are in our ability to think? I don’t know and neither do you but it’s a question worth considering.

Scott 149: “Consider that birds must have some notion of fluid dynamics to fly”
Do I have to have a notion of electrodynamics when I switch on the light? Maybe more to the point: Do children have to have a notion of grammar when they learn speaking? Maybe you are saying that the birds (or the children’s) nervous system must encode something equivalent to a physical (or logical) theory and this representation must be accurate enough to work in the real world. Maybe this representation IS mathematics???

A H 01.28.14 at 6:22 pm @ 165:‘… Well, there are two ways that people talk about mental behavior. Sometimes by a thought they mean a unit of subjective experience and sometimes they mean a unit of private linguistic behavior. So if you mean the first definition, then you can have thoughts for which there are no words, but if it is the second you can’t. … Since mathematics is a linguistic phenomena….’

I have heard some mathematicians very strongly object to the idea that mathematics is linguistic. I myself have had the experience of attempting to solve problems where the first perception of the problem and its possible solution seemed to be geometrical (loosely speaking) rather than verbal. I’ve also had this experience in playing certain games (chess, go), in computer programming, and in artistic work. It could be, of course, that some sort of linguistic engine was running out of conscious sight and somehow presenting things as spatial or tactile concepts, but if so it has certainly stayed out of sight. There could also be other forms of non-linguistic thought, for example whatever music is and does — although it can be reduced to sequences of symbols I don’t think that’s the way J.S. Bach or Thelonious Monk made it up.

You might, as a matter of convention, select a particular group of rules. But it is not the case that because the set of rules you pick is conventional, the truth of whether or not you are following the rules must also be conventional.

Well, as Armando has said, maybe “formal” is a better way to understand mathematical truth. But, seems to me you’re incorrect. If you make a mistake while trying to follow the rules then you have made a formal or conventional mistake.

If the math is somehow “inaccurate”, then our descriptions of the world must also be inaccurate. … It may be the case that there’s some way to replace all of the math in all of our theories with something else that does just as well without being equivalent to math.

AFAIK we do NOT have a tight mathematical model for the universe. Far from it. So there is plenty of room for improvement. But I was not suggesting that our “math” is faulty, only that there is a good chance that new mathematical ideas and systems will eventually be used to describe difficult-to-model phenomena, of which there is plenty. And when that happens what does that imply about the earlier math? Are you denying that Newtonian physics (and the math associated with it) was shown by Einstein to be only an approximation of a subtler reality?

But it is not the case that because the set of rules you pick is conventional, the truth of whether or not you are following the rules must also be conventional.

mattski:

Well, as Armando has said, maybe “formal” is a better way to understand mathematical truth. But, seems to me you’re incorrect. If you make a mistake while trying to follow the rules then you have made a formal or conventional mistake.

Well, first be clear about what I said. You can claim that the set of rules you pick is just a matter of convention. (It clearly is somewhat a matter of convention, but as the case of Armando’s plus vs. regular plus should make clear, there may be reasons that some conventions are preferable to others–and those reasons may not be arbitrary.) You can claim that whether or not you are following the rules is a matter of convention. But the second claim isn’t the same as the first.

Secondly, Armando should pay attention, because what you just said makes my point clear. You’re taking an anti-realism position much more extreme than formalism–that not only are the rules a matter of convention, but the implications of a given set of rules are a matter of convention. You aren’t the only anti-realist who doesn’t take logical consistency as a given truth–Lee Smolin in Time Reborn makes similar claims, and I think some other posters above would as well. Some interpretations of Wittgenstein on rules might point in that direction?

Are you denying that Newtonian physics (and the math associated with it) was shown by Einstein to be only an approximation of a subtler reality?

On the contrary, that’s my point–assuming Einstein was right, it is a fact of the physical world that his math fits the world closer than Newton’s math. I’m not claiming that math can’t be wrong–my math is wrong all the time–I’m claiming that the math can’t be wrong without the science also being in some sense wrong or incomplete.

What does this mean? How do you know when a set of rules has been “completely” described?

Armando gave a complete description of his “plus”: “add the next number plus one”. To completely describe a rule is to… completely describe it, just like in the directions for a board game: state when the rule applies and what to do when the rule applies. An unreal rule would be one you described only indirectly but nothing fit the description (e.g. “the rule which, when encoded as an integer and doubled is equal to this odd number”–sorry, can’t come up with a less artificial example, but none of my points hinge on a concept of “unreal rule”.)

What is math then?

My belief is just that when people apply the process typically called “math”, there is something human-independent about that activity, at least some of the time. I’m not claiming to know what that something is, and furthermore the “right” definition of “math” probably depends on what that is, so giving firmer definitions isn’t appropriate.

A H 01.28.14 at 11:37 pm @ 189:‘…@186 Besides a few diagrams, math is just words. When you see an equation on the page you can speak it out loud.’

Yes, and when I look at a Bach fugue printed on a page, I can say ‘d, f, a-flat’ and so on, but something’s missing both coming (how it got here) and going (what it does when it’s performed). The fact that something can be (partially) converted into language doesn’t mean it is language. Or at least, I don’t experience things that way.

You can claim that the set of rules you pick is just a matter of convention.

I think I see where you and I are not connecting. So… I never said that mathematical truth is just a matter of convention. Of course not! What I did say was that we ‘devised mathematics to describe the world.’ Not arbitrary at all. We created this language of mathematics for the express purpose of being useful.

But, similar to regular language, it is still a matter of convention that x+y=z. Same with “roses are red.” So I think you overstated my position quite a bit.

On the contrary, that’s my point–assuming Einstein was right, it is a fact of the physical world that his math fits the world closer than Newton’s math.

Wait. I thought it was my point! Our mathematical formulations are approximations which–we hope–are approaching reality. But, they are not reality! They approach reality as a limit. Are you sure you want to call an approximation “real” or “true”?

My belief is just that when people apply the process typically called “math”, there is something human-independent about that activity, at least some of the time.

I find it helpful to think how other animals might deal with this, as it avoids the language trap. Would it help to say that number is a category, that categories relate to the purposes of the categoriser (as noted at 171 above), and as you have more brain you can perform more complex processes with categories. So the raven has a process or idea of discrete units (so many other ravens/squirrels…) and some ability to perform operations with this idea – maybe add and subtract units up to some limit.

The point is that this idea – discrete units, and related ideas of more/less/the same as, and some basic operations (this number plus/minus this number equals this number), even if always conceived in concrete terms (ravens, squirrels, cats…) seems to relate so closely to the existing states of reality as to be both useful and mind-independently true. Even if you need a certain degree of mind to apprehend it at all, and some other degree of mind to apprehend more complex developments. In short, number could be unarbitrarily true, but maths an increasingly arbitrary extension of number.

Anon @33 had a good point: I can’t for the life of me convince myself that the question “are numbers real?” has a clear meaning or whether the commitment either way leads to interesting further commitments or consequences … What are the stakes of such a debate? What do we lose if numbers aren’t real?

Have been reading along hopefully, but I don’t see that anyone has answered, or even addressed his questions. (Well, maybe jch.)

In the same spirit, I’d ask: do we have an agreed-upon meaning for “exist”? One of my dictionaries defines it as “to have being or actuality of a material or spiritual nature.” The other defines it as “to be; to have reality or actual being.” As far as I can tell, these definitions are circular, hence utterly useless.

When I was a devout Catholic long ago, I made the mistake of beginning to study Scholastic philosophy. I was stopped cold by “Being.” I couldn’t make any sense of the notion, as metaphysicians use it. Then I came across the logical positivist/pragmatist dictum: “Faced with a metaphysical question, ask: What depends on the answer? How would the world be different if one answer or the other were true?” For the (eternal) life of me, I couldn’t see how the world would be different if “essences” or “substances” or “souls” or “numbers” existed in (metaphysical) reality, rather than as counters in a language game. I still can’t.

Sorry if I misstated your position. There are other people who hold that position. I can’t figure out how what you call “convention” corresponds to what I (and I think most people) call “convention”, but it doesn’t seem fruitful to dig any further.

Are you sure you want to call an approximation “real” or “true”?

I’m very sure that I want to call the fact that it is (or is not) a good approximation true. The correspondence (or lack thereof) between a real object and it’s a approximate model is a fact about the real object.

And I see no justification for such a belief.

The dependence of current physics and chemistry on mathematics. If math is not human-independent, then our current physical laws aren’t either.

Sure Consmatopia, but in addition to being a Platonist about math, I’m an Early Wittgensteinian. That’s just how I roll. Bring the crystalline logical totalizing structures. That of necessity fail on contact with our fallen world, sadly, but.

If you hold that mathematical expressions are (more or less) just like other nominative expressions, you’re either going to have to hold a radically anti-realist line, or you’re going to pretty much concede something very much like Platonism. “Roses are red” is a complicated example, but take “gold is a metal”. Obviously, the signs ‘gold’ and ‘metal’ depend on certain conventions, but we use to pick out these conventionally defined signs to pick out aspects of independently existing reality. So, gold e.g. is an independently existing kind of thing that we refer to using the conventionally defined symbol ‘gold’. Now suppose for the sake of argument that our category of ‘metal’ is inexact and will in the future be replaced by a more exact category—this future category will be more exact because it’ll more accurately pick out the independently existing kind of thing that we now refer to using ‘metal’.

Now if mathematical expressions are just like that, then ‘2’ functions just like ‘gold’ or ‘metal’ does—it refers to aspects of independently existing reality. Of course, ‘2’, ‘two’, etc. are conventionally defined, and of course they refer, if they do, to abstract entities and not to material objects or kinds of objects, (tho kinds may be abstract entities anyway, but just skip that), and moreover it won’t matter if ‘2’ etc. are inexact somehow and will be replaced by expressions that do a better of job of categorizing whatever it is that we use numerical expressions to pick out now.

This isn’t really my view, by the way. It’s just that if there is an important distinction between the way mathematical expressions function and the way in which expressions designating spatiotemporal objects function, (as I do think there is), then this distinction can’t turn on how exact or approximate these expressions are.

Wittgenstein is surely right about the interpretation of rules, though. Kant actually makes exactly the same point in the first Critique. (Near the beginning of the Analytic of Principles, in the unlikely chance that anyone cares.) I don’t think this commits either of them to any bizarrely implausible form of conventionalism. (Surely no one thinks Kant’s too much of a conventionalist?)

Mathematics depends on the SAME cognitive operations as physics and chemistry: Seeing things particulate or separated, carrying a memory of one to the next, finding connections between them, etc. The necessary operation is that the lifeworld splits into two (Plotinus and LEJ Brouwer).

Humans could have math but with a very different physics and chemistry. In an alternate universe. If an alternate universe might have a different set of physical constants (as we now suppose is possible), then why not a different set of physics equations? Go the whole hog!

But an alien race, even in this universe, which exists as colored lights that hum to each other might not have our metaphysics of seeing particles in forcefields. They might not have the necessary intra-cognitive splitting. They could be extraordinarily advanced in understanding the universe, but they might not do math, physics or chemistry.

Perhaps stars like our Sun are conscious. They only feel other stars, they are only conscious to other stars. (They don’t bother with their planets; we are motes in their eyes!) In all their lightbeams, time is stopped dead (that is OUR physics) — thus, they communicate instantaneously. All stars in the universe are in instantaneous communication; maybe they compose one big creature. Okay then: It is grander than you. Does it do math or physics?

Whales have bigger brains than humans, and appear (to me) to be more intelligent. There is evidence that their language is complex and memorous (or memory-laden). Do numbers exist for whales?

I think there’s something obvious that hasn’t been mentioned, unless it’s in one of the many many many comments I haven’t read, and that’s that probably some of the resistance to talking about “Platonism” comes from all the stupid crap associated with it. That and liking to argue on the internet.

We just don’t know what the neural basis of subjective consciousness, willful action, or language is. So it is hard to say exactly what the difference is between sensory experience and linguistics acts, because at a fundamental level we don’t even know where these mental phenomena come from.[…] it is a genuine mystery how this happens, though I am optimistic it will be solved within the next 50 years or so.

It seems unlikely that we’ll have a thinker with no sensory input for comparative purposes even by then, though.

We just don’t know what the neural basis of subjective consciousness, willful action, or language is.
Not just that, our designated thinkers-about-these-things don’t seem to have picked up a lot of humility on the way to that not knowing.
A hundred plus years max of even the bare ground of microbial awareness and where’s the “Wow! We thought we knew pretty much what was up on the local scene and it turns out an invisible reality was saturating our lives all along, and we had no idea it was there.
Here.”

Didn’t even know that what we are ourselves to start with is just another tiny thing on that microscopic scale.
And the whole time it was that way and the whole time until just yesterday we didn’t know anything about it except how to make beer and bread.
So what’s the odds on more to come? At that level of shift. Of awareness.
What’s the likelihood that if there is as Lee Arnold suggests, a kind of sentience in the stellar nucleus, it’s so far beyond our grasp it might as well not be there.
Not someone you could converse with, or make demands of, but exactly what you get when energy of that magnitude exists in one place, which it does.

I can’t separate wheat from chaff in sci-journalism, but Sir Roger Penrose is evidently pretty chuffed about being right about something microtubules quantum something brain. Thought.

Up and down don’t really exist. They’re just a local convenience. Like numbers.
Vital, essential, and they disappear completely around ten miles up, as well as just over the horizon.
Not to mention the non-theoretical center of the earth. Try to get “down” working there.

It’s hard for some of us to imagine reality without mathematical essence, it’s like describing “a box with no inside” or making a drawing of a Klein bottle.
A word game, not a weirdness at the heart of things. Or a glory.
But then some people can’t imagine thinking without words, or signs. And others can’t imagine not being able to do that.

“Secondly, Armando should pay attention, because what you just said makes my point clear. You’re taking an anti-realism position much more extreme than formalism–that not only are the rules a matter of convention, but the implications of a given set of rules are a matter of convention.”

Actually, the rules of implication are up for grabs. For reals. Not only could you see the axiom of choice like this – what mathematical objects can we work with – but also more fundamentally in things like accepting or rejecting the law of the excluded middle.

It is also important to note that if you execute a set of rules on a computer – eg arithmetic – then there will be things about those objects which the computer cannot see. If this is one’s criterion for reality, would that mean that there are facts about integers which are neither true nor false?

Is the weird kid at school doing mathematics when she annoys her teacher by getting the correct answer while being unable to explain how? I’m not sure that a purely language-based system encompasses all of what I would want to include.

Re: the Hellenistic tradition, I think I’m right in saying that they had a concept of number that was restricted to integers. Rational numbers would have been considered as something else (ratios but not numbers), and irrationals simply as magnitudes, again strictly not numbers. And of course they did lots of interesting and correct mathematics, just not all the kinds of things we can do using a broader definition. (I’m using ‘we’ in a broad sense obviously – I’m hopeless).

I also wanted to clarify why I find Consumatopia’s position not very convincing.

So suppose one asks whether works of art are created or discovered. (I don’t think anyone really asks this, so perhaps it is a little unfair, but I think one could outline a Platonic position that works of art all exist in some other realm/in potentiality and all we do is discover them.) OK, so thats a question.

Now someone comes along and says, of course Casablanca exists and because I can save it on my computer and play it back, it is clearly human-independent. I’ve got a computer to play it, after all! And you see that no one would really find that convincing, since the issue is not whether or not I can watch a film called Casablanca on a computer. No one is debating that, and it has no bearing on whether the thing was discovered or invented. It demonstrates a reality that no one opposes, so doesn’t say very much.

armando 01.29.14 at 11:14 am @ 212 — Some have contended that works of art are discovered rather than created, or are made unknowingly as a hen makes an egg, arising like Venus from the sea of the (collective?) unconscious. Michelangelo famously said that the sculpture already existed in the stone, and all you had to do was cut the extraneous material away — although maybe he was joking. Many artists have the experience of ‘seeing’ the work whole before they begin making it. To others the vision grows, exfoliates, falls into place over time. In the case of a movie, however, the details of modern collective industrial production would mostly obscure these original impulses.

Now, now, john c. halasz, are you certain about the prices? I need have only one item of infinite value in the set and the whole thing’s an incredible bargain relative to cost, you know.

godoggo: if it wasn’t for arguing on the internet where would we all be? After all?

In general: should I express mild surprise for having successfully trolled everyone with alleged doctrinaire adherence to the actual Platonism of Plato in particular? I don’t actually think that thing, so that’s why it has not been the means by which I trolled everyone. Not am I lying about my own views in order to troll the blog; I actually think the thing I claim, about math. However, I am not able to articulate a good defense, so I just made an assertion. Mainly, other people had preëxisting desires to argue about math and I gave them the chance.

It ought to be pointed-out that the use of the word “Platonism” in philosophy of math is very well-known and very specific. Usually called “mathematical Platonism” or “ontological realism”, i.e. the belief that numbers and other mathematical objects exist independently of human minds. There are some famous mathematical Platonists: Gödel was mentioned above. Roger Penrose has written repeatedly that the Mandelbrot set existed prior to discovery.

The dependence of current physics and chemistry on mathematics. If math is not human-independent, then our current physical laws aren’t either.

Yes, I agree. Our current ‘laws of nature’ are human constructs. I may be wrong but my sense is that most scientists would agree that they are not “laws” in the commonly understood sense. They are statistical approximations that we base our current efforts on. A scientist, of all people, is more likely than most to allow that it is possible for a rock to roll UP a hill. In fact, quantum mechanics sees no violation of nature in such a scenario. But it is deemed exceedingly unlikely.

I hope to come back to the thread later today. Meantime, per arguing on the internet, it is always wise to call to mind the words of the great Primo Carnera,

Actually, the rules of implication are up for grabs. For reals. Not only could you see the axiom of choice like this – what mathematical objects can we work with – but also more fundamentally in things like accepting or rejecting the law of the excluded middle.

You can use a constructive logic that doesn’t hold either of those, but still derive statements conditional on them being true. Better examples would be substructural logics, in which weakening, contraction and/or exchange rules are not permitted. (If you’re using linear logic without exponentials, you can’t put something on the left of an implication that lets you do unlimited contractions on the right.) Still, even if different logics derive different statements, there should still be a fact of the matter as to which statements are derivable in each logic (if there isn’t then we have to take exactly the anti-realist position you claim no one is taking).

It is also important to note that if you execute a set of rules on a computer – eg arithmetic – then there will be things about those objects which the computer cannot see. If this is one’s criterion for reality, would that mean that there are facts about integers which are neither true nor false?

I don’t know if I’d go so far as to say that only semidecidable properties have a “fact of the matter”, but the further you climb up the arithmetical hierarchy the more in doubt that is. I think it’s exactly that kind of doubt that makes it reasonable to talk about whether a mathematical truth “exists” or not–not all of them necessarily do.

It demonstrates a reality that no one opposes, so doesn’t say very much.

Well, no, I think some people do oppose it. Some people have a much more radical anti-realist position than you do. I was badly mistaken in assigning that view to mattski, and I guess there’s doubt as to whether even late Wittgenstein believed that (I’m no expert), but I still stand by the statement that this is not a strawman, even if I’m not sure exactly who’s putting it forward ;-).

That aside, I think the possibility of implementing rules in machines has more relevance to the question of whether math is discovered than art is discovered because mathematics is inherently about the behavior of rules, while Casablanca is not at all about the behavior of video decompression algorithms (unless I misunderstood it completely!)

@120 Belle: “in addition to being a Platonist about math, I’m an Early Wittgensteinian.”

Can one be both? Oddly, my initially posted aversion to the question of real math was inspired by early Wittgenstein. As I see it, W’s view is that propositions make pictures of real *relationships among* objects, but that this tells us nothing about the nature or reality of the *objects* depicted.

Mathematical propositions are true descriptions of real states of affairs, but this doesn’t tells us whether numbers are “real.” Wittgenstein suggests that in order to depict facts, language must treat objects *as if* they were simple objects or reducible to simple objects (which would be the ontological basis of “real” number), but insists this doesn’t prove there are simple objects–hence the “ladder” ending.

Language, in effect, works by picturing real relationships between bits of the world *as we’ve carved it up by language*. So number is an effect of language, not a pregiven reality. But it’s far from arbitrary because, given any linguistic mesh laid over reality, the relations between the delineated objects will be consistent, determined by the real structure of the world.

It’s not that math isn’t “true” or doesn’t depict reality, but that the “reality” it depicts is not of “things” but of relations.

I’m pretty sure it is a strawman, Consumatopia, and that the position you are describing as “realist” is more commonly known as “anti-realist”. This makes the exchange with you a bit odd, to say the least. But, in any case, we agree that one can write down the rules for a formal system and more or less decide certain facts about it.

I’ve enjoyed the pastime of seeing what meta mash people will try to make of mathematics, but in some ways I am left wanting more.

The power of math is a bit mysterious, no? Does it depend on notation? I think it kind of does, but why is that so? If notation is tied directly to human cognitive incapacities, does that shape math itself? Is notation neutral? Neutral in relation to what?

Is the demand that math be “real” just a tribute to its surprising power? The kids think nothing is real until it is on teevee, maybe math isn’t real until it’s on YouTube.

And, finally, let me reiterate my previous complaint: too many are way to eager to leap to an uncritical assertion about math being “descriptive”. What is the relation of math to poetry? Or drama? Or comedy? Or rhetoric?

“A scientist, of all people, is more likely than most to allow that it is possible for a rock to roll UP a hill.”

But that position is itself based on mathematical theory (namely, statistics). It actually expresses confidence that current math-based physical theory is a good description of reality.

“Is the demand that math be “real” just a tribute to its surprising power?”

I’m still intrigued by the case of the birds having a “notion” of fluid dynamics, which to my regret wasn’t further discussed. I have thought more about this and it occurred to me that the ear must have a “notion” of Fourier analysis. What should we think of this: does the ear actually do Fourier analysis (i. e. mathematical manipulation without a conscious human mind directing it), or is Fourier analysis just a mathematical model we have invented to describe what the ear (and similar phenomena) do? I realize this is wishi-washi and needs to be fleshed out more but perhaps this has been done already? Is there any philosophical tradition that takes this viewpoint seriously?

“But, in any case, we agree that one can write down the rules for a formal system and more or less decide certain facts about it.”

I don’t think everyone here would agree that there are facts about formal systems that are true independent of our deciding that they are true.

Also, the position I described goes a bit further than “more or less decide certain facts about them”. In particular, I would say that there’s a fact of the matter as to whether a theory is consistent, even if deriving that consistency within the theory itself would prove it inconsistent.

My kids already thought this shit; I merely elicited it, a la Socrates and the slave boy, but with less slavery. But John and I agreed that each view is open to as many fatal objections as the others. If the numbers are in some meaningful sense outside of us, by what means do we interact with them, the pineal gland? Pencils? If the numbers are arbitrary human constructs, then how come people keep finding things out about astrophysics with math and the same numbers keep turning up and why would it be like that if there weren’t math? And so on?

Just one last flailing before I let it be. For those who want to claim that the “realism” of mathematics is legitimated by its scientific applications, it’s possible, often enough, that the mathematics of a theory can be written up in several different ways, or even expressed in non-mathematical, or at least non-quantitative terms, at least in part. For example, (from my very limited repertoire), cellular automata might be considered computational, but they’re not quantitative, and Feynman diagrams in quantum mechanics are sheerly pictoral, yet so inuitively clear that they’ve been adopted as standard representation for what otherwise would be expressed in complex equations. For that matter, Einstein had to learn (be tutored in) Riemannian geometry in order to fully work out and express general relativity, (so IOW the “intuition” preceded the math). In general, scientific theories depend on a working out of a framework of concepts, (identifying and interpreting their specific “object domain”), before evidence can be sought out and developed to support or confirm the theories, and competing theories are to be judged on the basis of how well their conceptual frameworks organize the relevant data or evidence, which latter are not completely theory-independent. IOW scientific theories are never themselves completely and simply empirical and reducible to observable elements requiring “ontological” commitments. So mathematical means or elements, (which are always non-empirical, based on prior abstraction of empirical experience), would be no different. (I also think Bruce Wilder is correct when he objects to the notion that math “describes” anything; it is used to formulate explanations, not descriptions). Further, “pure” mathematicians sometimes complain that the advanced math used by advanced physics is not quite right, that it’s kludgy, to which physicists respond, yeah, but it works.

Also, to reiterate a point I already made above, two completely distinct domain-specific theories might use both a mathematical formalism that is completely homologous, but nevertheless are encoding fundamentally different domain-specific concepts. Let’s call that the “identity of indiscernibles” trap, which does real damage to understanding. As, for example, when economic systems and physical systems are regarded as if they operate on the same level, because, say, an Euler equation determines optimal flows. And that does practical damage as well, as when literally rocket scientists are hired to design complex financial derivatives, (since they are the ones with the mathematical chops to do so), but they haven’t a clue as to the actual economic function or purpose of the derivatives they design, (until they blow up).

So the stakes in mathematical “Platonism” are not just gamey.

Also, it’s correctly claimed that Goedel was himself inclined toward such “Platonism”, even if his own theorem seems to cut against it. But then he was a very crazy fellow, (sharply inclined toward paranoia), and word has it that in later years he and Einstein used to take long walks, loopily discussing general relativity in terms of the reversibility of time and the attainment of a timeless universe. In the meanwhile, that Belgian priest used the same theory to derive the “big bang”, imparting to the physical universe its own temporal evolution.

Apologies. I would like to respond to you in a more considered way but today is shot. So just this:

If you hold that mathematical expressions are (more or less) just like other nominative expressions, you’re either going to have to hold a radically anti-realist line, or you’re going to pretty much concede something very much like Platonism.

I don’t understand what you are trying to get across. I don’t understand what position you are imputing to me.

I’m still intrigued by the case of the birds having a “notion” of fluid dynamics, which to my regret wasn’t further discussed. I have thought more about this and it occurred to me that the ear must have a “notion” of Fourier analysis. What should we think of this: does the ear actually do Fourier analysis (i. e. mathematical manipulation without a conscious human mind directing it), or is Fourier analysis just a mathematical model we have invented to describe what the ear (and similar phenomena) do? I realize this is wishi-washi and needs to be fleshed out more but perhaps this has been done already? Is there any philosophical tradition that takes this viewpoint seriously?

I don’t buy it, but his credentials as mathematician and philosopher certainly beat mine. I balk at step one of his proof, “Every physical entity is a computation.” It seems to be a metaphor stretched too far, like earlier scientists imagining every physical entity as a clock or a steam engine. Is this mathematical Platonism taken to a new level? Not only do humans discover math that was there all along, humans are math.

If the numbers are in some meaningful sense outside of us, by what means do we interact with them, the pineal gland? Pencils?

But they aren’t *only* outside of us, in (my version of) this view; they are inherent in everything, including whatever we happen to be thinking with, and whatever squid or aliens or pure asensory AIs think with. They’re here for us to feel out, although we (and, I hope, squid and aliens etc etc) will feel out different aspects.

I don’t think this makes me a panpsychic, but I can go along with `Every physical entity is doing a computation.’ Sadly, some of them seem to be doing it faster than the simulation I’m trying to code.

Hans George Gadamer once remarked that in the modern age, superstitions are as likely to be derived from (misapprehension of) modern science as from the obduracy of traditions. It’s amazing to me how few people here seem to have any conception of reification, or just the “fallacy of misplaced concreteness”. If modern science is to be an enhancement of our understanding, rather than just an overburdening of it, then it’s important to have an understanding of what science is and does and what it isn’t and doesn’t accomplish.

William Kahan’s advice is to never design a program on your best day, because you need your best day to test and debug it. I don’t know that I could work that up to a theology, but certainly there’s a good origin tale or so in it, with or without panpsychism.

Mathematics is useful because something like the Strong Church-Turing thesis (the universe is isomorphic to a computable process) is true–once you’re capable of simulating a universal turing machine (incredibly simple), you can simulate anything.

That does not answer Belle/John’s objection to Platonism–how does the eternal reality of math force the Strong Church-Turing thesis to appear to be true? In fact, I guess you could use it against Platonism–if Church-Turing explains everything, why bother with Plato? I guess the Platonist would counter that Church-Turing explains why there would be some math that explains the world, but not why that math is as simple or beautiful as they claim it is in the case of physics. But oddly enough, I suspect that the strongest opponents of Platonism will also oppose SCT.

While we’re asking how Platonic objects affect the world, we should note that some of the more prominent mathematical Platonists in physics also have non-reductionist views on qualia.

jch may ultimately be right, but Einstein’s intuitions prior to formalism are a point in favor of Platonism (listen to Penrose talk about intuition–or Hamming linked to above), as are Gödel’s incompleteness theorems (every formalized system of math has true statements that cannot be deduced within that system). Gödel believed many eccentric things, including extreme paranoia that eventually took his life, but his mathematical philosophy was serious, and it was probably his strong belief in the reality of the arithmetic that led him to consider the Gödel numbering his theorems rely on.

He rejects the notion that a concrete physical object in the universe can be ascribed a simple spatial or temporal extension, that is, without reference of its relations to other spatial or temporal extensions.

Well, okay, but there are relations between objects, so we’re good — no, apparently they aren’t good enough to build on. Why not?… The definition of extension (why not `extent’?) takes us to the Planck length and Spinoza, but not the initiation of the Big Bang, which seems like a missed opportunity.

I was ready to say that I subscribe to the Strong Church-Turing thesis, but how does it deal with quantum randomness? If I believe that the half life of an ensemble of radon atoms is computable, is that sufficient, or must I believe that the veil of quantum randomness can eventually be lifted and the remaining lifetime of a single radon atom is at least in-principle computable?

Really, Lee, is that how you interpret Whitehead? I would think that emergent evolution and the conception of an open-ended universe was more his point, a non-reductive physicalism, not the thoughts of rocks. It’s the contemporary Analytic philosopher Galen Strawson who spouting on about “pan-psychism”, precisely because be doesn’t think emergentism can be causally made out.

Whitehead explicitly stated that he was writing in support of American pragmatism, but, as it were, to save it from itself, presumably meaning its lack of theoretical reflection, its anti-intellectual tendencies. And his formulation of the “fallacy of misplaced concreteness” was intended to defend and protect the practical concerns of human beings from the arbitrary impositions of science and its tendency toward sheer instrumentalism. ‘Cause there is such a thing as ordinary experience, knowledge and understanding “in the life-world” and its relation to science is not a one-way street. Scientific theories have to make sense, since if something is senseless, meaningless, then how could it be true? It’s a matter of a “fusion of horizons” between the two.

It’s an old philosophical question/dispute as to whether only particulars are real or whether relations are too. I myself (and Whitehead) lean toward the latter. But relations don’t simply cash out as mathematical computations.

If the numbers are arbitrary human constructs, then how come people keep finding things out about astrophysics with math and the same numbers keep turning up and why would it be like that if there weren’t math?

Back to my question about whether we can conceive ratios without numbers. If the universe is ‘folded’ in such and such a way why wouldn’t we see these patterns reflected in our numbers?

“If the numbers are in some meaningful sense outside of us, by what means do we interact with them, the pineal gland? Pencils?”

Yes but that problem isn’t restricted to what we think of as mathematical entities. Because the Sakura problem hasn’t been solved. We don’t “interact” with the flower, we interact with a mental image that our nervous system has formed that we recognize as flower. That’s not saying that we make the flower up. In a realist account, the mental flower is related to objective reality but the mental image cannot be identified with that reality. Our mental image has properties such as shape, color and number of petals, which correlate to aspects of objective reality, and there is no fundamental difference between the way our mind makes use of the concept of color versus the concept of number.

@248 — John you are probably right, because I always make the mistake of thinking that “panpsychism” includes developed varieties such as Leibnitz’ “monads” and Whitehead’s “actualities” with vectorial “prehensions” — things which (to me) skirt the borders of consciousness. I suppose that is wrong, and panpsychism should be kept as the simple label for “thoughts of rocks”?

And you are certainly right, insofar as Whitehead was a theist, and God puts Whitehead’s process-metaphysics into motion. So I imagine that Whitehead would have taken a very strong objection to hearing his philosophy called “panpsychism”.

Matt 236: “Every physical entity is a computation.” “Not only do humans discover math that was there all along, humans are math.”

*Mathematics as simulation*

I’m more comfortable to say that physical entities are simulated by computation/Math (which is not the same or maybe it is?). We invented Fourier analysis in order to simulate empirical entities like the ear. We invented numbers to simulate the behavior of heaps of pebbles and the like (and discovered that the same simulation that works for pebbles also works for coins, etc.) We invented geometry to simulate physical space. We invented calculus to simulate dynamic systems. And so on. That at least gives an account for the “unreasonable effectiveness”. This goes against the grain of picturing Math as a terribly abstract domain but I do think most Math started out as a simulation of empirical phenomena. Then, much later, the simulation itself became object of investigation, and higher order simulations were invented, and things got so abstract one can now pretend that Math is separate from empirical science. I think that view is not just mistaken but also damaging to students who are trained in that attitude.

Nothing so far answers the question why mathematical simulation of empirical reality is possible. I think what makes it possible is consistency in the way the Universe works. We could perhaps imagine a Universe without consistency and not only would there be no Math, there would be no intelligent beings in such a Universe because there would be no such thing as information (maybe I’m being narrow-minded, maybe one could imagine intelligent beings that do not exchange information but I just can’t). One could interpret this as a form of anthropic reasoning: there is Math because without Math, we wouldn’t be here.

I wrote, “A scientist, of all people, is more likely than most to allow that it is possible for a rock to roll UP a hill.”

TM responded, But that position is itself based on mathematical theory (namely, statistics). It actually expresses confidence that current math-based physical theory is a good description of reality.

First of all, it isn’t based solely on theory. It is also based on an attitude of agnosticism about what is and isn’t possible. You could say, the ability to refrain from making absolutist statements.

Second, I don’t see how statistical means of understanding the physical world strengthens the case for mathematical platonism. Especially since fairly simple visual means are available to make statistical statements without mathematics.

Really, statistics strikes me as a very modest form ‘knowledge.’ And from what we know (?!) about quantum mechanics the physical world rests on a foundation of scarcely knowable chaos. Or so it seems to me.

I’m not sure what your point is mattski. I wasn’t defending platonism. I don’t know what it means “make statistical statements without mathematics”. I don’t think its agnosticism to say that a stone could, theoretically, spontaneously roll upwards. This theoretical possibility is an implication of a specific model (matter is composed of atoms which move around randomly, so it is just conceivable that by chance they all move in the same direction.) The physicist isn’t saying, I’m agnostic about whether matter is composed of atoms.

I don’t know what it means “make statistical statements without mathematics”

I am sure you have seen a pie chart.

This theoretical possibility is an implication of a specific model (matter is composed of atoms which move around randomly, so it is just conceivable that by chance they all move in the same direction.)

So, only a person who believes this specific model can also believe that a rock could possibly roll up hill?

The physicist isn’t saying, I’m agnostic about whether matter is composed of atoms.

“Suppose that just this second the Universe began behaving somewhat inconsistently.”

Nice question. But now we have to specify what is meant by inconsistent. My argument would probably have to lead me to say that inconsistency in the sense I mean the term would preclude our continued existence, so nobody would be here to worry about prime numbers.

mattski: I guess we don’t agree on terminology. I don’t think we could have pie charts and not have mathematics.
“So, only a person who believes this specific model can also believe that a rock could possibly roll up hill?” Nah, that’s nothing to do with my argument. I think there’s a difference between saying that A is predicted to be possible by scientific theory X, and A could happen because of magic. Again, we may disagree that these positions are different.

engels 231: I always wonder when reading articles like that: what happened next? Did somebody come up with a refutation or was it just left hanging there?

It is no refutation of platonism to do math with visual illustrations and intuition–that’s exactly the kind of math platonists like most.

@TM, how about as a definition of consistent: “follows some finitely describable rule”? That’s not what consistent means in other contexts, but I think it captures what makes mathematics useful. A universe in which that weren’t at least locally true is one in which living beings would never think of mathematics, whether or not they could exist at all.

I don’t understand what you are trying to get across. I don’t understand what position you are imputing to me.

mattski,

I wasn’t so much imputing a position to you as presenting you with a dilemma (sort of). Briefly: if you want to say that (a) numbers and math more generally is a set of heuristic devices we’ve invented because it helps us explain certain aspects of the world, and that (b) things metals, roses, etc., do indeed exist in the world independently of anything going on with us humans, then you don’t want to say that (c) our ‘math language’ let’s say functions more or less just like the language we use to designate spatiotemporal objects—or that out ‘math language’ and our ‘object language’ is conventional in the same way. Asserting (a), (b), and (c) together can quickly get one into trouble that needs a lot of fancy footwork to avoid.

Two other things:

1. Platonists obviously don’t think that numbers, e.g., are spatiotemporal entities! So asking “where” they would be is quite besides the point. Rather, the claim is that we can and should include certain kinds of abstract entities in our ontology—since we’re talking about abstract entities they’ll of course lack spatiotemporal properties like location. Now you might think that this is weird enough (and I e.g. think that the epistemic problem Belle mentions is pretty fatal to the view), but you don’t want to strawman the platonist.

The intuitive pull of Platonism for some derives from what happens when we ask questions like, “if no one were around to add 2 and 2, would 2 still exist as a number and would 2+2 still be equivalent to 4″?

And what happens, of course, is that we view the world as if no one is there – but of course we’re the ones viewing it – so we’re there. We can’t imagine a world without us, in some sense.

Nonetheless, when we think we are imagining a world without us, then it seems perfectly plausible that 2+2=4 even when all human beings have turned to dust. After all, when we ask whether 2 would still exist in such a world, we’re not looking for it visually – we’re really asking whether it makes sense to think of the number 2 in such a world. And it does of course, because we’re doing the thinking.

But frankly I’m not clear on what an abstract object is, or for that matter what a thought really is. I don’t think anyone is clear on those things. And I think you’d need to be clear on those things to settle on what mathematical Platonism is about (if anything at all).

How about this question:

You happen to be there as a badly damaged spacecraft skims to a landing on a frozen river in northern Maine. As it coasts to a stop, the hull splits open, and a creature resembling an odd mix of dolphin and human staggers out, his uniform ripped and bloodied. In one hand, the gray skin blistered with burns, he clutches what appears to be a book. Collapsing into the snow in front of you, he hoarsely tells you in perfect Icelandic that he is the last of his crew and his species; and that the book contains the history of his species and an important warning. But, he gasps, the language is very unlike any human language, and is heavily encrypted as well. Before you can make any inquiries, he places the enormous book into your gloved hands, and dies.

Curious, you open the book, and find remarkably that whenever your glance falls upon a series of symbols not separated from one another by a space, the book emits a certain collection of sounds, some barely audible.

Do any concepts contained in the book that are unknown to human beings exist at this moment? Do they exist only if decryption and translation are logically possible? Do they exist only if decryption and translation are actually possible? Do they exist only if decryption and translation are actually effected?

If one views mathematics as a description of the physical world, then there are some similarities between those questions and the one in this thread.

So instead of the old ontological worry about whether anything is “real”… we have a full-fledged speculative realist ontology, in which nothing is illusory, but everything is ultimately inaccessible. This seems to me to be right and accurate. Dick’s novels (think of 3 Stigmata or Ubik) show how Descartes’ ontological disquiet is thoroughly “naturalized” or “objectified” in modern (mid-20th-century) commodity capitalism. But I think that this structure has entirely imploded in our current neoliberal world: instead of a Dickian sense of unreality as a result hypercommodification, we realize — or we are forced to accept — that such commodification itself is entirely real (a “real abstraction” — abstraction itself is the most concrete thing we can experience), along with the way that “interiority” is now restructured as “human capital,” in “investing” which we are forced to be entrepreneurs of ourselves.

Is this trolling? What are the differences, social and personal, between the sense of the “real” in the above and the one discussed in this thread? Do I have to understand both Whitehead and Baudrillard and codeswitch between the two? Is that even possible?

The difference, I think, is a personal commitment to ontology as social and political engagement.

2nd Thesis on Feuerbach

“The question whether objective truth can be attributed to human thinking is not a question of theory but is a practical question. Man must prove the truth — i.e. the reality and power, the this-sidedness of his thinking in practice. The dispute over the reality or non-reality of thinking that is isolated from practice is a purely scholastic question. “

How about a picture of several apples, some of one color and some of a different color?

Nah, that’s nothing to do with my argument.

I get the feeling you are indifferent to the context of my original statement. But I’ll offer you this, if an event occurred which seemed to violate our current “laws” of nature–a rock rolling up hill to take an unlikely example–you can bet that scientists would look for an explanation while suspending judgement. Magic would not be invoked.

Thank you. So, I didn’t mean to suggest that language and mathematics are conventional in the same way. If I gave that impression it was more due to my own laziness in trying to keep my remarks brief. I was considering offering the view that–excluding higher mathematics–is it not uncontroversial to say that the “truths” of arithmetic are tautological in nature. Whereas it is not at all obvious that “roses are red” has that same quality. To take an example.

Dear john c. halasz: it’s amazing to me that it seems to you that it’s amazing how few people here seem to have any conception of reification, or just the “fallacy of misplaced concreteness”. Because, just move your thoughts to probabilities for a moment and consider: what are the odds I have never thought of this, ever?

“How about a picture of several apples, some of one color and some of a different color?”

In a specific sense, this IS mathematics, or let’s say concepts that we think of as mathematical (set, number) are implicit in that picture. I’m still not sure what your point is though. It seems to me that your position is sort of conventionalist and you think I am platonist but I am actually realist. I see math neither as existing independently of empirical reality nor as something we make up.

I said above that for mathematics to be conceivable, the Universe must be “consistent” in some nontrivial sense. To spell that out a bit, a minimum requirement for that consistency is that there are physical entities that can be in distinct states, and that there is some regularity in the way these states change. That for example is a minimum requirement for information to be physically representable and transmittable. Without that, there could be no life, let alone intelligence. Now as soon as there is a system capable of representing distinguishable states, the primitive set concept arises. In that sense, “set” is an aspect of physical reality. An intelligent being trying to simulate (i. e. mentally represent) physical entities cannot do it otherwise than by employing the concept we call set – whether or not they come up with a language for that, and whether or not this happens in some sense consciously.

But physical reality doesn’t contain an entity that is a “set”. It contains entities with distinct states that can be represented as sets. Likewise numbers, and so on.

I have to add something here because what we think of as “systems” and “states” is in fact not simply empirically given but already a result of some classification. And that is perhaps what really makes thinking about this so difficult. Again, the red five-petaled flower is presented to our mind by our visual system after hugely complex neuronal processing. It’s not just there – it’s how we see it. Yet how we see it is not arbitrary or made up, otherwise our species would have gone extinct long ago. It’s almost impossible to even describe this precisely. Or maybe it is impossible.

The comment was made in reference to all people ‘here’, i.e. in the thread.

As Belle suggests, you just need to think about probability. If it is truly amazing, in your eyes, the number of people ‘here’ who have no concept of reification, then what are the odds that Belle, in your eyes, has the concept?

Obviously you actually have no reason to suppose Belle lacks the concept. But, by the same token, you have no reason to suppose anyone else here lacks the concept. Yet you are supposing this thing. So it seems reasonable for Belle to suppose that, even though you have no reason to suppose she lacks this concept, you are indeed supposing she does so. Your words give her a reason to suppose you suppose it, though you lack a reason to do so.

In a specific sense, this IS mathematics, or let’s say concepts that we think of as mathematical (set, number) are implicit in that picture.

-A picture of many apples
-A picture of a large circle next to a small circle
-A picture of a small circle inside a large circle
-A bowling ball placed on a table next to a golf ball.
-A table upon which are placed many apples.

Are you saying these are “mathematics?” If so, I think you are speaking your own private language. There is also the sense that you’re jerking my chain. I hope that isn’t the case.

What I have argued is tat we have developed a language of sets and numbers because physical reality is such that sets and numbers are built into it. So when you describe “A table upon which are placed many apples”, you are describing a set, whether you do it consciously or not, and you couldn’t perform that description without a set concept represented somewhere in your nervous system. And to say that by describing apples on a table you are employing mathematical concepts is not an abuse of language, like deciding to call a chair an apple. It is not far-fetched to say that mathematics starts from grouping things together, classifying them, identifying relations between them. And these things happened unconsciously long before the human mind evolved.

Look, I earlier suggested that the ear could be said to conduct Fourier analysis. I. e. math, or simulation, happens in physical reality without an intelligent mind doing it consciously (and that is a hint to understanding the “unreasonable effectiveness of mathematics”). You may say you don’t buy that and that’s fine with me but is the point I’m trying to make really so obscure?

and you couldn’t perform that description without a set concept represented somewhere in your nervous system

Do you have evidence for this assertion?

And to say that by describing apples on a table you are employing mathematical concepts is not an abuse of language

That would depend heavily on the nature of the description, if you ask me. (Leave aside that I was pointing here at scenarios rather than descriptions of scenarios.) But I don’t see how you write what you have written after saying @ 256 that you “weren’t defending platonism” unless you niggle over definitions to the point of absurdity.

It was claimed that birds when they fly are doing aerodynamic calculations. (Why is the birds doing the calculating, rather than the air?)

Or it was claimed that when we perceive a flower, we’re really just perceiving a mental image of a flower. (The reason our perceptual and other categories match up with one another is that, aside from being a common species, we exchange symbolic tokens with each other when we communicate across the world. And we can generally handle the difference between perceiving something real and perceiving just an image of it).

In general, one shouldn’t confuse a concept with the reality to which it refers. (That’s a tricky business, since all human experience, activity, knowledge and understanding is mediated by our meanings and concepts, and one can’t somehow strip them of the world and compare them directly either to the world or our experience of it). But many here seem to have assumed that because science offers a unique, privileged access to the underlying reality of the world, scientific concepts are immediately and intuitively referent, (when, in fact, we must always already have such access to the world in order to at all do science, and using scientific concepts to explain scientific concepts is rather question begging, no?) Perhaps the untranslatable Verstehen/Verstand distinction is one I should have mentioned as something many commenters seemed to overlook.

To repeat myself tediously, mathematical formalisms usually originate independently of empirical natural science and many such constructs have no scientific uses. and some legitimate branches of natural science make little use of such formalisms, (e.g paleontological taxonomy). So the argument that because math is used in science and science is referent to reality, math is intrinsic to the universe somehow and thus “ontologically” real, isn’t warranted. It applies to our measurements of that reality and why just those parameters or variables should be so measured requires a conceptual interpretation or “justification” and not just a mathematical one. The most Bell Waring has actually argued is that the natural world has real causal processes, (actually, I would say, many levels of such processes), and those processes have structures that are potentially intelligible, often by means of mathematical applications. I don’t disagree with that, but it does nothing to underwrite mathematical “Platonism”.

Further, I suggested that math, beyond the simplest operations, is likely not doable without symbolism, (though mathematical symbolism, which refers to pure formal operations, is not the same as linguistic symbolism and likely involves different thought processes with a different neural substrate). But then symbols are at once a re-duplication of reality and a “nihilation” of it. Insisting that their “ontological” status is that of reality, (independent of the very complex rules systems that constitute them), just strikes me as odd. And the claim that physics is the one real and true scientific account of the world and makes the world strikes me as having more a “theological” flavor than a scientific one. (It would be more accurate to say that far-from-equilibrium, turbulent thermo-dynamics makes the world as we know it, but how calculable is that?) I’m not drinking the Quinean kool-aide, which I think just ends up as a weirdly scientistic form of relativism.

And I offered a consequential example in the application of the Euler condition. Since Euler was not just a seminal, but a prolific mathematician, there are several so named, but it’s the one that maintains the alignment of the variables determining the optimal flow of a fluid. But are the flows of a physical fluid the same as monetary “flows”? It’s a piece of math required to “close” an economic model with rat-ex and inter-temporal optimization, not any actual empirical observation. (And I’m surely not the only one who thinks economists spend far to much time studying their models, rather than studying the world). Which models underwrite claims for “sound finance” rather than “functional finance”. The latter requires alternative economic theories of endogenous money and finance with debt dynamics. So there are real and not just intellectual consequences to these issues: it doesn’t boil down to the “identity of indiscernibles”.

So I’m glad that Belle Waring actually does possess the concept of reification. (Actually, there are several more-or-less complex accounts of it, with Whitehead’s “fallacy” being just an aspect of it). But maybe she could reflect on the many ways it can be applied.

But then why is John Holbo replying to my reply to Belle Waring’s comment that was addressed to me, other than for the sake of sheer bumptiousness? I am under the referential delusion that they are two separate persons.

“mathematical formalisms usually originate independently of empirical natural science and many such constructs have no scientific uses.”
At one end of things we have as a pressure against completeness what Godel said (yes, I know this is in a totally different direction than your comment – a moment please). But this just appears to be an argument against confidence that we can place or denote what actually undergirds mathematical truths. Yet, if we actually do have a mathematical system that consists of propositions that are considered true, and what is considered to be true is rigorously applied, then developments from those principles should remain true. The only unknown is whether our beliefs are really true, and whether our beliefs are too narrow or wide in comparison to the truth. But that is apparently just a mismatch between confidence in the completeness of our experience and in our willingness to extrapolate.

More simply, I think your comment implies that many mathematics which are considered promising, and produced formally, but turn out to be inaccurate (and, to my way of thinking, turning out not to be rigorous at all) have actually been rejected because of appeals to the truth. There is a lot of disagreement about what “empirical” might entail here so I will speak more vaguely about “being perceived as true” with the assumption that it is indeed true.

It would seem confusing, if not obviously incorrect, to say that math exists independently of empirical science. It might not be immediately obvious, or even testable, whether a particular mathematical construction would obtain (because it might assume a framework that doesn’t exist in the reality we know, and possibly that it also has no use as a “virtual” thing, such as imaginary numbers and models of unusual topologies sometimes seem to be of use), but that doesn’t mean that we have done anything other than to come into the process of grasping truth in the middle. A lack of knowledge of the actual foundations of things doesn’t imply a complete lack of knowledge, to my way of thinking – for me the situation is essentially exactly akin to the problems of foundationalism in knowledge and empiricism more generally.

I don’t take this as particular arguments for or against Platonism but it does seem to move in that direction.

If I hold to a metaphysical “-ism”, it is agnosticism. The easy way out! But really I don’t trust the arguments and judgments of people who insist that there IS a God, and I don’t trust the arguments and judgments of people who insist that there IS NO God. Rationality could not supply the answer either way, and there’s the end to it. Mathematical Platonism is much more interesting, but still I am agnostic, because how can you possibly know?

That said, I think mathematical Platonism is very easy to defend:

Causal processes have been investigated and sometimes it is found that quantities are conserved. To find that out, you have to count or measure. The number “2” is a placeholder for a position in the act of counting. Its meaning is different from “1” or “3”; “2” is after “1” and before “3”. But to say that numbers are therefore merely the placeholders in the communication of a measurement, in order to avoid saying that numbers exist Platonically, advances you nothing. You have already given that the world has real causal processes, even if no humans existed, and I am going to assume that you will allow that some quantities are conserved in some of those causal processes, even if no humans existed. So quantity exists Platonically. Quantity is number.

The Maxwell’s demon thought experiment–a tiny entity that sorts individual molecules to reduce entropy–was originally conceived to illustrate the statistical nature of the Second Law of thermodynamics. But then people started to ask exactly why such a tiny organism or machine would have to violate the laws of physics at the atomic scale. One suggestion: the entity would have to expend energy (tiny light signals were imagined) to measure the surrounding molecules in order to sort them, and this energy would have to be at least equal to the energy gained by sorting. That’s wrong, but a lot of people believed that (perhaps because people are too quick to want a paradox resolved, by analogy with an observation under the Uncertain Principle, or by analogy with then-popular Cybernetics in which information was inherently valuable (though I think that Norbert Wiener himself actually avoided this mistake) and decades later in the late 90s that’s still how the problem Maxwell’s Demon and it’s resolution were explained to me by very smart people. I’m sure a lot of people still believe it today.

It turns out that the laws of physics, as believe both back then and today, allow a mechanism to copy information (to measure some part of a molecule’s state, put another molecule in correlation with that state) with arbitrarily small increase in entropy. What stops Maxwell’s Demon from breaking the Second Law is Landauer’s principle: resetting one bit of information (dividing the possible number of microstates in the macrostate of the world by two) must increase entropy by at least kT ln 2, which is the most that the Demon could increase energy in any cycle. The time reversibility of the laws of physics (at the micro level, not at the level of daily experience) forces us to pay a price for forgetting information, not for learning it. (Whether or not that’s how reality works, the notable thing is that it’s how the theory works, and people were getting that wrong.)

It’s my suspicion that mistakes somewhat like this are what’s actually at the root of some of the more disastrous social failures of applied abstraction (not just the Wall Street quants, but everything in Seeing like a State). But I don’t have a full theory yet to back that up.

But here my disagreements begin. I would not accept your solution to this problem–that we never look for patterns across disparate domains. In particular, I think that the way we keep making mistakes across all these domains in similar ways probably means that there are generally applicable things to learn here. But I don’t think we’ve precisely explained what those ways are. It’s not just “Reification!”. Not every thought process that someone chooses to label reification is fallacious. Some of this is perfectly valid.

I don’t think the right lessons from Wall Street and the disasters enumerated in Seeing is “destroy all abstraction!” Better responses: beware of technocrats. Ban high-frequency trading. Increase capital requirements for banks. And dsquared is probably right that some kind of crisis was inevitable given monetary and fiscal policies of the major countries. And the finance-physicist apologism of The Physics of Wall-Street wasn’t totally convincing, but there were some good points there. Platonists might not even be the droids you’re looking for here. The Black Swan frequently claimed that Platonists and Bayesians had ruined everything–if only we’d listened to Mandelbrot! Thing is, I’m pretty sure that most platonists are more fond of Mandelbrot than Bayes.

More generally my view is don’t stop people from experimenting with radical ideas, stop people from ruining innocent lives with those experiments. You’ve given one side of the stakes. But not the other. A lot of physicists and mathematicians clearly have platonist inspirations–even if platonism is wrong, they might not have made the breakthroughs they did without believing it (or gone on to study the fields they studied at all). Gödel probably being the most prominent example of that. There are entire research programs that are called into question if mathematical anti-realism is true. (e.g. meta-mathematics and mathematical quasi-empiricism). I happen to think those programs look extremely promising, and it would take much more that the arguments you’ve presented here to convince me otherwise. I did not lack “any conception of reification”, nor do I think you’ve provided much evidence that anyone else in particular does. Just because someone disagrees with you and/or Whitehead doesn’t mean they haven’t heard of this stuff at all. You can conclude that we’re wrong, but you’ve got no basis to attribute as much ignorance as you did.

And the claim that physics is the one real and true scientific account of the world and makes the world

You definitely don’t have to make any reductive claim like that to be a mathematical realist or platonist of any sort. David Deutsch is platonist perhaps even more extreme than Tegmark, and he’s argued at length against a “theory of everything” (not necessarily the attempts to reconcile QM and relativity, but the reductivism implied by the phrase ). Physical reductionism underlies some of the mathematical anti-realist positions (see Lakoff and Núñez linked above). I find it a bit odd that you would accuse platonists, of all people, of having a “tendency toward sheer instrumentalism” or denying ordinary experience.

To repeat myself tediously, mathematical formalisms usually originate independently of empirical natural science and many such constructs have no scientific uses. and some legitimate branches of natural science make little use of such formalisms, (e.g paleontological taxonomy). So the argument that because math is used in science and science is referent to reality, math is intrinsic to the universe somehow and thus “ontologically” real, isn’t warranted.

No one has to deny that mathematical formalism are conceived for reasons other than modeling an empirical theory, that there is much math with little apparent value to science, or that some fields of science have more math than others, to argue for the “unreasonable effectiveness” of math in some sciences, or for mathematical realism or empiricism in general. In fact, I would say that math being useful some but not all of the time suggests that it’s apparent utility is not just a matter of everything looking like a nail when I hold a hammer.

Insisting that their “ontological” status is that of reality, (independent of the very complex rules systems that constitute them), just strikes me as odd.

The Löwenheim–Skolem theorem says that no formal theory of an infinitely sized mathematical model can fully specify which model its referring to–there are always some annoying non-standard models (like a set of natural numbers, but where some “number” is its own successor) sneaking in. Slightly more prosaically, by Gödel, any formal theory of mathematics must include some true statements that cannot be proven in the theory. I don’t expect that stuff to convince everyone, but maybe it might help you understand why it doesn’t strike everyone as oddly as it strikes you? To put it another way, what part of “Incompleteness” don’t you understand–the formalisms aren’t complete because there is always more math outside the formalism. (And it isn’t because mathematics is somehow “too big” to describe in a single formal system–it’s because doing so generates a contradiction.)

“But then why is John Holbo replying to my reply to Belle Waring’s comment that was addressed to me, other than for the sake of sheer bumptiousness?”

It’s perfectly normal to point out to people that they are being a bit rude, John. And it’s perfectly normal to point out to people that they are being a bit unreasonable. I don’t deny a motive of self-amusement. (Bumptiousness is not my style, however, as you well know.)

Let’s try to turn an etiquette point into a recipe for improved philosophy. Your contributions to this thread are no doubt thoughtful and serious. Thank you for that. But your signature style is dogmatic, and perhaps a more dialectical approach will suit the subject better. When someone disagrees with you, don’t just speculate wildly about how they lack basic concepts. If they are educated, and have studied philosophy, it probably isn’t that. Consider whether they might not have a point. Or half a point.

So let me propose an exercise (I do not say I set it for you – I am not your teacher, I know that). Write a forceful rebuttal to your own #275. Don’t fantasize that the opposition is going to make flagrantly stupid assumptions like, “because science offers a unique, privileged access to the underlying reality of the world, scientific concepts are immediately and intuitively referent.” (It isn’t very likely that people are going to fail to see the problem with that, is it?) Instead, see if you can imagine the opposition avoiding the most obvious pitfalls, yet without being driven to accept what you say.

Think about how your own use of ‘concept’ (for example).

Think about how this is going to provide the opposition with several avenues of attack and objection:

“mathematical formalisms usually originate independently of empirical natural science and many such constructs have no scientific uses. and some legitimate branches of natural science make little use of such formalisms, (e.g paleontological taxonomy). So the argument that because math is used in science and science is referent to reality, math is intrinsic to the universe somehow and thus “ontologically” real, isn’t warranted.”

This misrepresents the opposition and, what’s more, it encodes presuppositions that are likely to be held against you. In some ways you have their argument upside down and backwards. It is the very fact that math is sometimes done in an utterly non-empirical spirit (as you say) yet is then shown to have some empirical application (as you admit) that drives the ‘unreasonable effectiveness of mathematics’ line. I don’t by any means regard the simple argument along these lines as decisive, but you are not doing justice even to the simplest version. So obviously you aren’t refuting potentially more sophisticated developments of it.

You are bopping from point to point, opening vast fronts that invite skepticism and attack. “mathematical symbolism, which refers to pure formal operations, is not the same as linguistic symbolism and likely involves different thought processes with a different neural substrate”

How do you know that math symbolism (always?) refers to pure formal operations? Do you really regard it as likely that the neurology is so functionally bifurcated as you suggest? (Wouldn’t we have seen it already in the scans?)

You are going to object that this is just a comment box and you can’t fill in all the details, and guard against every misunderstanding. But then: do the opposition the courtesy of assuming the same. They, too, are limited by the comment box format. I’m not going to call you an idiot, just because it looks to me like several of your comments are open to quite elementary objections. Now you return the favor. (Get it?)

Your view is sophisticated and serious, but you don’t take seriously the thought that the opposition is also sophisticated and serious, not merely lacking in basic concepts. Philosophy of mathematics is marked by sophisticated and serious people saying things like what you say, but also by sophisticated and serious people who defend very different views. (It is also marked by sophisticated and serious people seriously straw-manning the opposition. You are in good and respectable company, no doubt. Still, a little of that goes a long way, surely.)

Very briefly, since I have to get into the bath and to bed, since I have to rise early tomorrow to attend , yes, a scientific conference (on the post-carbon future), though only to listen, not to present. I’m not opposing any formalism, whether pure or applied, nor any advancement of research. Nor does the notion of reification imply somehow the abolition of all abstracts. To the contrary, words are already generalizations and we are always already caught up in some level of abstraction. Rather the point is that abstractions can be both well or poorly formed or derived and well or poorly applied. And reifications needn’t be false to be mistaken. E.g. Ir’s actually true that we are just tubes of protoplasm. But outside some specialized contexts, it’s not useful or relevant to say so, since it entails a damaging eliminationism with respect to embodied persons. And the point about instrumentalism was not that “Platonists” endorse it, but rather that they tend to ignore its possibility, and I think an instrumentalist reading of science is damaging, since after all the aim is a kind of objective truth not instrumental power.

@282:

You are often bumptious, Holbo, even if it’s for your “self-amusement”. I offered a couple examples of what I considered reifying mistakes, without going through the whole thread, and my original remark was in passing, not referring to B.W. since she scarcely has made any argument here. (And the unifying thread of my remarks has be what difference does it make, what implications might follow, which I think a reasonable line of response.) ANd I said likely that math and words are different processes, since there are math prodigies and idiot savants and words are much more diverse and complicated, less consistent and “pure” than math and since the phenomenological experience of thinking in words, if implicitly, and doing math is much different. Nothing dogmatic about that statement.

jch, not wanting to step on John Holbo’s toes again because that’s heckuva rude and not the first time I’ve done that, but it is clear that I misread you somewhat and I apologize for that.

I would say, though, that some Platonists physicists and mathematicians I’ve read are quite aware of the possibility of instrumentalist readings of science and very clear that they don’t like it. But I haven’t taken a survey.

Possibly we have disagree about the meaning of the word ‘bumptious’. Google gives me “irritatingly self-assertive”. I would say that I am more irritatingly passive-aggressive. Surely those are not the same.

“ANd I said likely that math and words are different processes, since there are math prodigies and idiot savants and words are much more diverse and complicated, less consistent and “pure” than math and since the phenomenological experience of thinking in words, if implicitly, and doing math is much different.”

This all seems to me highly doubtful, hence doubtable (although if one takes ‘different process’ in a weak enough sense, some of it might pass as trivially true.) Certainly it is interesting that there are math prodigies – and music prodigies, one might add. But these phenomena are tantalizingly suggestive, given our current level of knowledge about the brain. What is more, they are ambiguously suggestive. Platonists spin the child prodigy phenomenon, and our sense of math’s purity, as evidence for their view. How are you so sure the tantalizing hints should be spun in your favor, not theirs?

As for phenomenology: that line seems to me hopeless, frankly. I don’t see that there is a characteristic phenomenology to mathematics or a characteristic phenomenology to ‘using words’ (vague phrase!) Rather, there is a wide range of phenomenological possibilities in both columns. Doing math may feel like nothing at all, just as ‘using words’ may feel like nothing at all. A math savant may have a synaesthetic phenomenology of mathematics (Daniel Tammet) that cannot be taken as a necessary concominant of mathematics, since most people who do math don’t share it.

Counting your change at the shop doesn’t ‘feel like’ trying to prove something in geometry. Trying to prove something in geometry may feel like trying to do some word puzzle. It feels like trying to make the pieces go where you want them, shuffling them around. Chatting with the man behind the counter, while you count your change, doesn’t ‘feel like’ reading or writing poetry. And on it goes. This isn’t solid enough ground that you can argue on this basis, I think.

You will say that phenomenology isn’t just ‘saying what you feel like’. I get that. But it doesn’t help, because pushing into the thicket of what phenomenology is, or ought to be about, just complicates matters. It doesn’t make things come clear in a way that supports your position.

Hey Consumatopia, thanks greatly for the mega-comment. I have a question about the abortive attempt to solve the Demon paradox: Given that we have Landauer’s limit, is the original thought experiment misleading in that it also leads people to expect that there are actually two steps to this sorting process, logically leading to an energy expenditure higher than that predicted by Landauer?

I also wonder if that has any relevance to the concepts of reversible computing. As far as I can tell, the assumption in Maxwell’s thought experiment that there are no external sources of disturbance (energy inputs), and that the walls of the containment are essentially inert as well, did not appear to lead people directly to the insight that you would want such a stable (i.e., low-energy) state of affairs as seems necessary for reversible computing, because of the Demon experiment’s apparent emphasis on maintaining energy levels (and possibly for some other reason/s), though my mind wonders why the Demon’s apparatus might not be set up in such a way that is allows the performance of calculation or some other useful task by setting up basic circuits with the gates. For example, I think that perhaps you could probabilistically control the directional flow of energy by setting multiple Demon-mediated traps into a wall, or within separate cells leading into the same cell, with traps leading the other way if desired.

Also think I see JCH’s point a bit clearly now with respect to eliminationism/reductivism – the problem apparently not being that we are stuck somewhere along the path of causality (from the foundations of knowledge) but rather that some reifications are totally separated from that path (untestable) while simultaneously appearing plausible.

Briefly, since I’m exhausted after a long day. I have a more “essential” reason, than “mere” phenomenology”, (since yes, I do think I have reasons for what I say, layers of reasons), and that is that math is sheerly digital, dependent on units, even if those units are infinitesimal points or some of the more exotic species of number theory, whereas natural language is both an analog and a digital “system”. And animal communication is sheerly analog. (If you don’t understand that distinction, ask Lee Arnold). So though I don’t think it’s quite right that math propositions are just sheerly tautological, they occur under the sign of identity and consistency, whereas linguistic expressions and their rules are are more complexly entangled and occur under the sign of difference and are involved in their uses and applications in making sense of the world of human experience and activity in a way that math is not. Especially language is necessarily ambiguous, which is anathema to math.

I also have probably a more relaxed sense of argument than you, since I don’t think arguments necessarily “prove” anything, or accomplish much, as opposed to offering some perhaps compellingly persuasive reasons to some, and I think that impasses are always possible. (Philosophic thinking since its inception has always been aporetic, puzzling, and it’s claims to “universality” doubtful, which isn’t to underwrite any radical skepticism).

As for bumptiousness, I thought it was a “truth universally acknowledged” in these part that academic philosophers are peculiarly prone to such a vice, with their “knock-down arguments” and knocking down of arguments and the like. You might think that your rope-a-dope tactics are merely “passive-aggressive”, but you might understand why such an “invitation” is rejected without being “rude” for declining the neutrality of all arguments, as if the only “commitments” were purely “logical” ones.

“As for bumptiousness, I thought it was a “truth universally acknowledged” in these part that academic philosophers are peculiarly prone to such a vice, with their “knock-down arguments” and knocking down of arguments and the like.”

Sorry, are you faulting me for not living down to these low standards?

My point was this: given that you hold your own arguments only to a loose and relaxed standard – nothing wrong with that, especially not on a blog – given that you think you have reasons you aren’t offering at the moment – nothing wrong with that – it is intellectually unfair and unhelpful to assume that those on the other side who are also arguing a bit loosely can only be doing so because they lack basic concepts, or are otherwise making some really obvious error. You would not like it if I told you that your relaxed arguments were indicative of lacking basic concepts. So you should extend the courtesy to the other side – not merely out of politeness, but because nothing intellectually good can come of doing otherwise.

As an example: you are right to presume that I don’t accept your digital-analog points. But you are wrong to presume that this is because I am unable to grasp the distinction between digital and analog.

Ergh! I forgot to mention the matter of “professional deformation”. And believe it or not, I read and am influenced in my thinking by much else besides philosophy. Nor is “reification” a basic concept; it is a conception of what can happen to “basic concepts”. Nor did I fail to extend “basic courtesy”, (as if that were the sine qua non of all internet traditions). Nor Is “intellectually good” some unique desideratum: there is such a thing as mistaken rigor. And it tends toward meaningless games of intellectual one-upmanship, which are less about “truth” than performance and the will-to-power. Nor did I impugn the different heuristics by which others operate. Rather, certain persons may have *chosen* to take offence, in the name of “courtesy”, for obscure reasons or motives of their own.

@Ed, to be honest, my information on the demon and thermodynamics only extends a little bit further than what I wrote up there. The demon and reversible computing are definitely related–Charles Bennett ties them together into a theory of the thermodynamics of computation.

But, motivated by your comment, I started researching it this evening and Landauer’s principle is more controversial than I realized. Apparently it’s possible to redesign the process so the entropy cost is paid during the measurement-like step rather than the reset-like step. And there’s a much more far-reaching criticism that in any reversible computer, thermal fluctuations would build up over time and eventually make the process as likely to move backwards as forwards.

Technology is apparently quite far from being able to build these things and test them, so the section of my mega-post related to all of this should be taken with a grain of salt.

Are you taking ‘Holbo is deformed’ as a premise, or a conclusion? Or neither. It makes a bit of a difference.

“Nor is “reification” a basic concept;”

In philosophy, the idea that you might mistake a substantive for a substance is pretty basic. It’s the 21st century, after all.

“Rather, certain persons may have *chosen* to take offence, in the name of “courtesy”, for obscure reasons or motives of their own.”

Let’s just grant that there is a certain amount of will-to-power and one-ups-manship on all sides. The question then becomes: might there be any intellectually sound reason for me to point out that you seem to be straw-manning the opposition? I think there obviously is. (Even if my motive is utterly bogus. I am a professionally deformed one-ups-man. Let it be so.)